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{{short description|Integer in the infinite Fibonacci sequence}}
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{{redirect|Fibonacci Sequence|the chamber ensemble|Fibonacci Sequence (ensemble)}}
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[[File:34*21-FibonacciBlocks.png|thumb|300px|right|A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.]]
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In mathematics, the '''Fibonacci numbers''', commonly denoted {{math|''F<sub>n</sub>''}}, form a [[integer sequence|sequence]], called the '''Fibonacci sequence''', such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,{{Sfn | Lucas | 1891 | p=3}}
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<math display=block>F_0=0,\quad F_1= 1,</math>
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<math display=block>F_n=F_{n-1} + F_{n-2}</math>
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for {{math|''n'' > 1}}.
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The beginning of the sequence is thus:<ref name=oeis>{{Cite OEIS|1=A000045}}</ref>
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<math display=block>0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots</math>
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Under some older definitions, the value <math>F_0 = 0</math> is omitted, so that the sequence starts with <math>F_1=F_2=1,</math> and the recurrence <math>F_n=F_{n-1} + F_{n-2}</math> is valid for {{math|''n'' > 2}}.{{Sfn | Beck | Geoghegan | 2010}}{{Sfn | Bóna | 2011 | p=180}} In his original definition, Fibonacci started the sequence with <math>F_1=1, F_2=2</math><ref>{{cite book |last1=Leonardo da Pisa |title=File:Liber abbaci magliab f124r.jpg - Wikimedia Commons |date=1202 |url=https://commons.wikimedia.org/wiki/File:Liber_abbaci_magliab_f124r.jpg |language=en}}</ref>[[File:FibonacciSpiral.svg|right|thumb|The Fibonacci spiral: an approximation of the [[golden spiral]] created by drawing [[circular arc]]s connecting the opposite corners of squares in the Fibonacci tiling; (see preceding image) ]]
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Fibonacci numbers are strongly related to the [[golden ratio]]: [[#Binet's formula|Binet's formula]] expresses the {{mvar|n}}th Fibonacci number in terms of {{mvar|n}} and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as {{mvar|n}} increases.
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Fibonacci numbers are named after the Italian mathematician Leonardo of Pisa, later known as [[Fibonacci]]. In his 1202 book ''[[Liber Abaci]]'', Fibonacci introduced the sequence to Western European mathematics,{{Sfn|Pisano|2002|pp=404–05}} although the sequence had been described earlier in [[Indian mathematics]],<ref name="GlobalScience" /><ref name="HistoriaMathematica" /><ref name="Donald Knuth 2006 50" /> as early as 200 BC in work by [[Pingala]] on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.
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Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''[[Fibonacci Quarterly]]''. Applications of Fibonacci numbers include computer algorithms such as the [[Fibonacci search technique]] and the [[Fibonacci heap]] data structure, and graphs called [[Fibonacci cube]]s used for interconnecting parallel and distributed systems.
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They also appear [[Patterns_in_nature#Spirals|in biological settings]], such as branching in trees, [[phyllotaxis|the arrangement of leaves on a stem]], the fruit sprouts of a [[pineapple]], the flowering of an [[artichoke]], an uncurling [[fern]], and the arrangement of a [[pine cone]]'s bracts.
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Fibonacci numbers are also closely related to [[Lucas number]]s <math>L_n</math>, in that the Fibonacci and Lucas numbers form a complementary pair of [[Lucas sequence]]s: <math>U_n(1,-1)=F_n</math> and <math>V_n(1,-1)=L_n</math>.
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== History ==
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{{see also|Golden ratio#History}}
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[[File:Thirteen ways of arranging long and short syllables in a cadence of length six.svg|thumb|Thirteen (''F''<sub>7</sub>) ways of arranging long (shown by the red tiles) and short syllables (shown by the grey squares) in a cadence of length six. Five (''F''<sub>5</sub>) end with a long syllable and eight (''F''<sub>6</sub>) end with a short syllable.]]
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The Fibonacci sequence appears in [[Indian mathematics]] in connection with [[Sanskrit prosody]], as pointed out by Parmanand Singh in 1986.<ref name="HistoriaMathematica">{{Citation|first=Parmanand|last=Singh|title= The So-called Fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|issue=3|pages=229–44|year=1985|doi = 10.1016/0315-0860(85)90021-7|doi-access=free}}</ref><ref name="knuth-v1">{{Citation|title=The Art of Computer Programming|volume=1|first=Donald|last=Knuth| author-link =Donald Knuth |publisher=Addison Wesley|year=1968|isbn=978-81-7758-754-8|url=https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100|page=100|quote=Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135&nbsp;AD) and Hemachandra (c.&nbsp;1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)...}}</ref>{{sfn|Livio|2003|p=197}} In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration {{math|''m''}} units is {{math|''F''<sub>''m'' + 1</sub>}}.<ref name="Donald Knuth 2006 50">{{Citation|title = The Art of Computer Programming | volume = 4. Generating All Trees – History of Combinatorial Generation | first = Donald | last = Knuth | author-link = Donald Knuth |publisher= Addison–Wesley |year= 2006 | isbn= 978-0-321-33570-8 | page = 50 | url= https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 | quote = it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ...there are exactly Fm+1 of them. For example the 21 sequences when ''m''&nbsp;=&nbsp;7 are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)}}</ref>
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Knowledge of the Fibonacci sequence was expressed as early as [[Pingala]] ({{circa}}&nbsp;450&nbsp;BC–200&nbsp;BC). Singh cites Pingala's cryptic formula ''misrau cha'' ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for {{math|''m''}} beats ({{math|''F''<sub>''m''+1</sub>}}) is obtained by adding one [S] to the {{math|''F''<sub>''m''</sub>}} cases and one [L] to the {{math|''F''<sub>''m''−1</sub>}} cases.<ref>{{Citation | last = Agrawala | first = VS | year = 1969 | title = ''Pāṇinikālīna Bhāratavarṣa'' (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan | quote = SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC}}</ref>
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[[Bharata Muni]] also expresses knowledge of the sequence in the ''[[Natya Shastra]]'' (c.&nbsp;100&nbsp;BC–c.&nbsp;350&nbsp;AD).<ref>{{cite journal|title=The So-called Fibonacci Numbers in Ancient and Medieval India|last=Singh|first=Parmanand|url=http://www.sfs.uni-tuebingen.de/~dg/sdarticle.pdf|journal=[[Historia Mathematica]]|year=1985|publisher=[[Academic Press]]|volume=12|issue=3|page=232|doi=10.1016/0315-0860(85)90021-7}}</ref><ref name=GlobalScience>{{Citation|title=Toward a Global Science|first=Susantha|last=Goonatilake|publisher=Indiana University Press|year=1998|page=126|isbn=978-0-253-33388-9|url=https://books.google.com/books?id=SI5ip95BbgEC&pg=PA126}}</ref>
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However, the clearest exposition of the sequence arises in the work of [[Virahanka]] (c.&nbsp;700 AD), whose own work is lost, but is available in a quotation by Gopala (c.&nbsp;1135):{{sfn|Livio|2003|p=197}}
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<blockquote>Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all ''mātrā-vṛttas'' [prosodic combinations].{{efn|"For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier&nbsp;– three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven [[Mora (linguistics)|morae]] [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <ref>{{Citation|last=Velankar|first=HD|year=1962|title='Vṛttajātisamuccaya' of kavi Virahanka|publisher=Rajasthan Oriental Research Institute|location=Jodhpur|page=101}}</ref>}}</blockquote>
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[[Hemachandra]] (c.&nbsp;1150) is credited with knowledge of the sequence as well,<ref name=GlobalScience/> writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."{{sfn|Livio|2003|p=197–98}}<ref>{{cite journal|last1=Shah|first1=Jayant|year=1991|title=A History of Piṅgala's Combinatorics|url=https://web.northeastern.edu/shah/papers/Pingala.pdf|journal=[[Northeastern University]]|page=41|access-date=4 January 2019}}</ref>
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[[File:Liber abbaci magliab f124r.jpg|thumb|A page of [[Fibonacci]]'s ''[[Liber Abaci]]'' from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.]]
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[[File:FibonacciRabbit.svg|thumb|The number of rabbit pairs form the Fibonacci sequence]]
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Outside India, the Fibonacci sequence first appears in the book ''[[Liber Abaci]]'' (1202) by [[Fibonacci]]{{Sfn|Pisano|2002|pp=404–05}}<ref>{{cite web|url=https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html|title=Fibonacci's Liber Abaci (Book of Calculation)|date=13 December 2009|website=[[The University of Utah]]|access-date=28 November 2018}}</ref> where it is used to calculate the growth of rabbit populations.<ref>{{Cite book|last=Hemenway|first=Priya|title=Divine Proportion: Phi In Art, Nature, and Science|year=2005|publisher=Sterling|location=New York|isbn=1-4027-3522-7|pages=20–21}}</ref><ref>{{cite web|url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits|title=The Fibonacci Numbers and Golden section in Nature – 1|last=Knott|first=Dr. Ron|date=25 September 2016|website=[[University of Surrey]]|access-date=27 November 2018}}</ref> Fibonacci considers the growth of an idealized (biologically unrealistic) [[rabbit]] population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?
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* At the end of the first month, they mate, but there is still only 1 pair.
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* At the end of the second month they produce a new pair, so there are 2 pairs in the field.
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* At the end of the third month, the original pair produce a second pair, but the second pair only mate without breeding, so there are 3 pairs in all.
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* At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.
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At the end of the {{math|''n''}}th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month {{math|''n'' – 2}}) plus the number of pairs alive last month (month {{math|''n'' – 1}}). The number in the {{math|''n''}}th month is the {{math|''n''}}th Fibonacci number.<ref>{{cite web | last = Knott | first = Ron
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| title = Fibonacci's Rabbits | url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[[University of Surrey]] Faculty of Engineering and Physical Sciences}}</ref>
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The name "Fibonacci sequence" was first used by the 19th-century number theorist [[Édouard Lucas]].<ref>{{Citation | first = Martin | last = Gardner | author-link = Martin Gardner |title=Mathematical Circus |publisher = The Mathematical Association of America |year=1996 |isbn= 978-0-88385-506-5 | quote = It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci | page = 153}}</ref>
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== Sequence properties ==
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The first 21 Fibonacci numbers {{math|''F<sub>n</sub>''}} are:<ref name=oeis/>
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:{| class="wikitable" style="text-align:right"
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|-
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| ''F''<sub>0</sub>
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| ''F''<sub>1</sub>
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| ''F''<sub>2</sub>
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| ''F''<sub>3</sub>
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| ''F''<sub>4</sub>
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| ''F''<sub>5</sub>
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| ''F''<sub>6</sub>
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| ''F''<sub>7</sub>
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| ''F''<sub>8</sub>
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| ''F''<sub>9</sub>
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| ''F''<sub>10</sub>
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| ''F''<sub>11</sub>
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| ''F''<sub>12</sub>
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| ''F''<sub>13</sub>
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| ''F''<sub>14</sub>
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| ''F''<sub>15</sub>
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| ''F''<sub>16</sub>
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| ''F''<sub>17</sub>
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| ''F''<sub>18</sub>
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| ''F''<sub>19</sub>
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| ''F''<sub>20</sub>
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|-
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| 0
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| 1
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| 1
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| 2
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| 3
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| 5
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| 8
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| 13
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| 21
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| 34
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| 55
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| 89
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| 144
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| 233
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| 377
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| 610
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| 987
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| 1597
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| 2584
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| 4181
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| 6765
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|}
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The sequence can also be extended to negative index {{math|''n''}} using the re-arranged recurrence relation
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<math display=block>F_{n-2} = F_n - F_{n-1},</math>
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{{anchor |Negafibonacci}} which yields the sequence of "negafibonacci" numbers<ref>{{Citation | last = Knuth | first = Donald | contribution = Negafibonacci Numbers and the Hyperbolic Plane | title = Annual meeting | publisher = The Mathematical Association of America | place = The Fairmont Hotel, San Jose, CA | date = 2008-12-11 | url = http://research.allacademic.com/meta/p206842_index.html}}</ref> satisfying
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<math display=block>F_{-n} = (-1)^{n+1} F_n.</math>
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Thus the bidirectional sequence is
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:{| class="wikitable" style="text-align:right"
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|-
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| ''F''<sub>−8</sub>
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| ''F''<sub>−7</sub>
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| ''F''<sub>−6</sub>
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| ''F''<sub>−5</sub>
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| ''F''<sub>−4</sub>
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| ''F''<sub>−3</sub>
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| ''F''<sub>−2</sub>
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| ''F''<sub>−1</sub>
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| ''F''<sub>0</sub>
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| ''F''<sub>1</sub>
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| ''F''<sub>2</sub>
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| ''F''<sub>3</sub>
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| ''F''<sub>4</sub>
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| ''F''<sub>5</sub>
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| ''F''<sub>6</sub>
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| ''F''<sub>7</sub>
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| ''F''<sub>8</sub>
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|-
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| −21
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| 13
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| −8
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| 5
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| −3
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| 2
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| −1
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| 1
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| 0
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| 1
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| 1
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| 2
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| 3
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| 5
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| 8
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| 13
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| 21
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|}
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== Relation to the golden ratio ==
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{{main|Golden ratio}}
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=== {{anchor|Binet's formula}}Closed-form expression ===
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Like every sequence defined by a [[Constant-recursive sequence|linear recurrence with constant coefficients]], the Fibonacci numbers have a [[closed form expression]]. It has become known as '''Binet's formula''', named after French mathematician [[Jacques Philippe Marie Binet]], though it was already known by [[Abraham de Moivre]] and [[Daniel Bernoulli]]:<ref>{{MathWorld|title=Binet's Fibonacci Number Formula|urlname=BinetsFibonacciNumberFormula}}</ref>
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<math display=block>F_n = \frac{\varphi^n-\psi^n}{\varphi-\psi} = \frac{\varphi^n-\psi^n}{\sqrt 5},</math>
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<math display=block>\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots</math>
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is the [[golden ratio]] ({{OEIS2C|id=A001622}}), and{{Sfn | Ball | 2003 | p = 156}}
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<math display=block>\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots.</math>
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Since <math>\psi = -\varphi^{-1}</math>, this formula can also be written as
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<math display=block>F_n = \frac{\varphi^n-(-\varphi)^{-n}}{\sqrt 5} = \frac{\varphi^n-(-\varphi)^{-n}}{2 \varphi - 1}.</math>
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To see this,{{Sfn | Ball | 2003 | pp = 155–6}} note that {{mvar|φ}} and {{mvar|ψ}} are both solutions of the equations
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<math display=block>x^2 = x + 1 \quad\text{and}\quad x^n = x^{n-1} + x^{n-2},</math>
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so the powers of {{mvar|φ}} and {{mvar|ψ}} satisfy the Fibonacci recursion. In other words,
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<math display=block>\varphi^n = \varphi^{n-1} + \varphi^{n-2}</math>
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and
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<math display=block>\psi^n = \psi^{n-1} + \psi^{n-2}.</math>
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It follows that for any values {{math|''a''}} and {{math|''b''}}, the sequence defined by
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<math display=block>U_n=a \varphi^n + b \psi^n</math>
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satisfies the same recurrence
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<math display=block>U_n=a \varphi^{n-1} + b \psi^{n-1} + a \varphi^{n-2} + b \psi^{n-2} = U_{n-1} + U_{n-2}.</math>
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If {{math|''a''}} and {{math|''b''}} are chosen so that {{math|1=''U''<sub>0</sub> = 0}} and {{math|1=''U''<sub>1</sub> = 1}} then the resulting sequence {{math|''U''<sub>''n''</sub>}} must be the Fibonacci sequence. This is the same as requiring {{math|''a''}} and {{math|''b''}} satisfy the system of equations:
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<math display=block>\left\{\begin{array}{l} a + b = 0\\ \varphi a + \psi b = 1\end{array}\right.</math>
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which has solution
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<math display=block>a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad b = -a,</math>
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producing the required formula.
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Taking the starting values {{math|''U''<sub>0</sub>}} and {{math|''U''<sub>1</sub>}} to be arbitrary constants, a more general solution is:
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<math display=block> U_n=a\varphi^n+b\psi^n </math>
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where
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<math display=block> a=\frac{U_1-U_0\psi}{\sqrt 5}</math>
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<math display=block> b=\frac{U_0\varphi-U_1}{\sqrt 5}.</math>
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=== Computation by rounding ===
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Since
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<math display=block>\left|\frac{\psi^{n}}{\sqrt 5}\right| < \frac{1}{2}</math>
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for all {{math|''n'' ≥ 0}}, the number {{math|''F''<sub>''n''</sub>}} is the closest integer to <math>\frac{\varphi^n}{\sqrt 5}</math>. Therefore, it can be found by [[rounding]], using the nearest integer function:
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<math display=block>F_n=\left[\frac{\varphi^n}{\sqrt 5}\right],\ n \geq 0.</math>
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In fact, the rounding error is very small, being less than 0.1 for {{math|''n'' ≥ 4}}, and less than 0.01 for {{math|''n'' ≥ 8}}.
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Fibonacci numbers can also be computed by [[truncation]], in terms of the [[floor function]]:
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<math display=block>F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\right\rfloor,\ n \geq 0.</math>
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As the floor function is [[monotonic]], the latter formula can be inverted for finding the index {{math|''n''(''F'')}} of the largest Fibonacci number that is not greater than a [[real number]] {{math|''F'' > 1}}:
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<math display=block>n(F) = \left\lfloor \log_\varphi \left(F\cdot\sqrt{5} + \frac{1}{2}\right) \right\rfloor,</math>
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where <math>\log_\varphi(x) = \ln(x)/\ln(\varphi) = \log_{10}(x)/\log_{10}(\varphi).</math>
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=== Limit of consecutive quotients ===
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[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio <math>\varphi\colon</math><ref>{{Citation|last=Kepler |first=Johannes |title=A New Year Gift: On Hexagonal Snow |year=1966 |isbn=978-0-19-858120-8 |publisher=Oxford University Press |page= 92}}</ref><ref>{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}</ref>
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<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>
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This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio, <math>-1/\varphi.</math>{{clarify|reason=What is a "pair" in -1/phi?|date=January 2019}} This can be verified using [[#Binet's formula|Binet's formula]]. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
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:[[File:Fibonacci tiling of the plane and approximation to Golden Ratio.gif|thumb|upright=2.2|left|Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous]]
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{{Clear}}
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=== Decomposition of powers ===
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Since the golden ratio satisfies the equation
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<math display=block>\varphi^2 = \varphi + 1,</math>
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this expression can be used to decompose higher powers <math>\varphi^n</math> as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of <math>\varphi</math> and 1. The resulting [[recurrence relation]]ships yield Fibonacci numbers as the linear coefficients:
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<math display=block>\varphi^n = F_n\varphi + F_{n-1}.</math>
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This equation can be proved by [[Mathematical induction|induction]] on ''n''.
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This expression is also true for ''n'' < 1 if the Fibonacci sequence ''F<sub>n</sub>'' is [[Generalizations of Fibonacci numbers#Extension to negative integers|extended to negative integers]] using the Fibonacci rule <math>F_n = F_{n-1} + F_{n-2}.</math>
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== Matrix form ==
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A 2-dimensional system of linear [[difference equations]] that describes the Fibonacci sequence is
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<math display=block> {F_{k+2} \choose F_{k+1}} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}} </math>
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<math display=block> \vec F_{k+1} = \mathbf{A} \vec F_{k},</math>
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which yields <math>\vec F_{n} = \mathbf{A}^n \vec F_{0}</math>. The [[eigenvalue]]s of the matrix {{math|'''A'''}} are <math>\varphi=\frac12(1+\sqrt5)</math> and <math>-\varphi^{-1}=\frac12(1-\sqrt5)</math> corresponding to the respective [[eigenvector]]s
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<math display=block>\vec \mu={\varphi \choose 1}</math>
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and
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<math display=block>\vec\nu={-\varphi^{-1} \choose 1}.</math>
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As the initial value is
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<math display=block>\vec F_0={1 \choose 0}=\frac{1}{\sqrt{5}}\vec{\mu}-\frac{1}{\sqrt{5}}\vec{\nu},</math>
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it follows that the {{mvar|n}}th term is
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<math display=block>\begin{align}\vec F_{n} &= \frac{1}{\sqrt{5}}A^n\vec\mu-\frac{1}{\sqrt{5}}A^n\vec\nu \\
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&= \frac{1}{\sqrt{5}}\varphi^n\vec\mu-\frac{1}{\sqrt{5}}(-\varphi)^{-n}\vec\nu~\\
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& =\cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^n{\varphi \choose 1}-\cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^n{-\varphi^{-1}\choose 1},
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\end{align}</math>
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From this, the {{mvar|n}}th element in the Fibonacci series
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may be read off directly as a [[closed-form expression]]:
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<math display=block>F_{n} = \cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^n.</math>
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Equivalently, the same computation may performed by [[Diagonalizable matrix#Diagonalization|diagonalization]] of {{math|'''A'''}} through use of its [[eigendecomposition]]:
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<math display=block>\begin{align} A & = S\Lambda S^{-1} ,\\
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A^n & = S\Lambda^n S^{-1},
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\end{align}</math>
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where <math>\Lambda=\begin{pmatrix} \varphi & 0 \\ 0 & -\varphi^{-1} \end{pmatrix}</math> and <math>S=\begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}.</math>
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The closed-form expression for the {{mvar|n}}th element in the Fibonacci series is therefore given by
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<math display=block>\begin{align} {F_{n+1} \choose F_{n}} & = A^{n} {F_1 \choose F_0} \\
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& = S \Lambda^n S^{-1} {F_1 \choose F_0} \\
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& = S \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} S^{-1} {F_1 \choose F_0} \\
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& = \begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}
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\begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix}
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\frac{1}{\sqrt{5}}\begin{pmatrix} 1 & \varphi^{-1} \\ -1 & \varphi \end{pmatrix} {1 \choose 0},
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\end{align}</math>
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which again yields
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<math display=block>F_{n} = \cfrac{\varphi^n-(-\varphi)^{-n}}{\sqrt{5}}.</math>
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The matrix {{math|'''A'''}} has a [[determinant]] of −1, and thus it is a 2×2 [[unimodular matrix]].
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This property can be understood in terms of the [[continued fraction]] representation for the golden ratio:
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<math display=block>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.</math>
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The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for {{mvar|φ}}, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:
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<math display=block>\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.</math>
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Taking the determinant of both sides of this equation yields [[Cassini's identity]],
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<math display=block>(-1)^n = F_{n+1}F_{n-1} - F_n^2.</math>
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Moreover, since {{math| '''A'''<sup>''n''</sup> '''A'''<sup>''m''</sup> {{=}} '''A'''<sup>''n''+''m''</sup>}} for any square matrix {{math|'''A'''}}, the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing {{math|''n''}} into {{math|''n'' + 1}}),
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<math display=block>\begin{align}
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{F_m}{F_n} + {F_{m-1}}{F_{n-1}} &= F_{m+n-1},\\
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F_{m} F_{n+1} + F_{m-1} F_n &= F_{m+n} .
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\end{align}</math>
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In particular, with {{math|1=''m'' = ''n''}},
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<math display=block>\begin{align}
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F_{2n-1} &= F_n^2 + F_{n-1}^2\\
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F_{2n} &= (F_{n-1}+F_{n+1})F_n\\
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&= (2F_{n-1}+F_n)F_n .
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\end{align}</math>
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These last two identities provide a way to compute Fibonacci numbers [[Recursion (computer science)|recursively]] in {{math|''O''(log(''n''))}} arithmetic operations and in time {{math|''O''(''M''(''n'')&nbsp;log(''n''))}}, where {{math|''M''(''n'')}} is the time for the multiplication of two numbers of {{math|''n''}} digits. This matches the time for computing the {{math|''n''}}th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with [[memoization]]).<ref>{{citation|title=In honour of Fibonacci|first=Edsger W.|last=Dijkstra|author-link=Edsger W. Dijkstra|year=1978|url=http://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF}}</ref>
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== Identification ==
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The question may arise whether a positive integer ''x'' is a Fibonacci number. This is true if and only if at least one of <math>5x^2+4</math> or <math>5x^2-4</math> is a [[Square number|perfect square]].<ref>{{Citation | title = Fibonacci is a Square | last1 = Gessel | first1 = Ira | journal = [[The Fibonacci Quarterly]] | volume = 10 | issue = 4 | pages = 417–19 |date=October 1972 | url = http://www.fq.math.ca/Scanned/10-4/advanced10-4.pdf | access-date = April 11, 2012 }}</ref> This is because Binet's formula [[#Binet's formula|above]] can be rearranged to give
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<math display=block>n = \log_{\varphi}\left(\frac{F_n\sqrt{5} + \sqrt{5F_n^2 \pm 4}}{2}\right),</math>
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which allows one to find the position in the sequence of a given Fibonacci number.
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This formula must return an integer for all ''n'', so the radical expression must be an integer (otherwise the [[logarithm]] does not even return a rational number).
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== Combinatorial identities ==
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Most identities involving Fibonacci numbers can be proved using [[combinatorial proof|combinatorial arguments]] using the fact that ''F''<sub>''n''</sub> can be interpreted as the number of sequences of 1s and 2s that sum to ''n''&nbsp;−&nbsp;1. This can be taken as the definition of ''F''<sub>''n''</sub>, with the convention that ''F''<sub>0</sub>&nbsp;=&nbsp;0, meaning no sum adds up to −1, and that ''F''<sub>1</sub>&nbsp;=&nbsp;1, meaning the empty sum "adds up" to 0. Here, the order of the summand matters. For example, 1&nbsp;+&nbsp;2 and 2&nbsp;+&nbsp;1 are considered two different sums.
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For example, the recurrence relation
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: <math>F_{n} = F_{n-1} + F_{n-2},</math>
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or in words, the ''n''th Fibonacci number is the sum of the previous two Fibonacci numbers, may be shown by dividing the ''F''<sub>''n''</sub> sums of 1s and 2s that add to ''n''&nbsp;−&nbsp;1 into two non-overlapping groups. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. In the first group the remaining terms add to ''n''&nbsp;−&nbsp;2, so it has ''F''<sub>''n''-1</sub> sums, and in the second group the remaining terms add to ''n''&nbsp;−&nbsp;3, so there are ''F''<sub>''n''−2</sub> sums. So there are a total of ''F''<sub>''n''−1</sub> + ''F''<sub>''n''−2</sub> sums altogether, showing this is equal to ''F''<sub>''n''</sub>.
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Similarly, it may be shown that the sum of the first Fibonacci numbers up to the ''n''th is equal to the (''n'' + 2)-nd Fibonacci number minus 1.{{Sfn | Lucas | 1891 | p = 4}} In symbols:
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<math display=block>\sum_{i=1}^n F_i = F_{n+2} - 1</math>
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This is done by dividing the sums adding to ''n''&nbsp;+&nbsp;1 in a different way, this time by the location of the first 2. Specifically, the first group consists of those sums that start with 2, the second group those that start 1&nbsp;+&nbsp;2, the third 1&nbsp;+&nbsp;1&nbsp;+&nbsp;2, and so on, until the last group, which consists of the single sum where only 1's are used. The number of sums in the first group is ''F''(''n''), ''F''(''n''&nbsp;−&nbsp;1) in the second group, and so on, with 1 sum in the last group. So the total number of sums is ''F''(''n'')&nbsp;+&nbsp;''F''(''n''&nbsp;−&nbsp;1)&nbsp;+&nbsp;...&nbsp;+&nbsp;''F''(1)&nbsp;+&nbsp;1 and therefore this quantity is equal to ''F''(''n''&nbsp;+&nbsp;2).
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A similar argument, grouping the sums by the position of the first 1 rather than the first&nbsp;2, gives two more identities:
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<math display=block>\sum_{i=0}^{n-1} F_{2i+1} = F_{2n}</math>
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and
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<math display=block>\sum_{i=1}^{n} F_{2i} = F_{2n+1}-1.</math>
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In words, the sum of the first Fibonacci numbers with odd index up to ''F''<sub>2''n''−1</sub> is the (2''n'')th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to ''F''<sub>2''n''</sub> is the (2''n''&nbsp;+&nbsp;1)th Fibonacci number minus&nbsp;1.<ref>{{Citation|title = Fibonacci Numbers |last1 = Vorobiev |first1 = Nikolaĭ Nikolaevich |first2 = Mircea|last2= Martin |publisher = Birkhäuser |year = 2002 |isbn = 978-3-7643-6135-8 |chapter=Chapter 1 |pages = 5–6}}</ref>
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A different trick may be used to prove
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<math display=block>\sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1},</math>
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or in words, the sum of the squares of the first Fibonacci numbers up to ''F''<sub>''n''</sub> is the product of the ''n''th and (''n''&nbsp;+&nbsp;1)th Fibonacci numbers. In this case Fibonacci rectangle of size ''F''<sub>''n''</sub> by ''F''(''n''&nbsp;+&nbsp;1) can be decomposed into squares of size ''F''<sub>''n''</sub>, ''F''<sub>''n''−1</sub>, and so on to ''F''<sub>1</sub>&nbsp;=&nbsp;1, from which the identity follows by comparing areas.
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=== Symbolic method ===
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The sequence <math>(F_n)_{n\in\mathbb N}</math> is also considered using the [[symbolic method (combinatorics)|symbolic method]].<ref>{{cite book |last1=Flajolet |first1=Philippe |last2=Sedgewick |first2=Robert |title=Analytic Combinatorics|title-link= Analytic Combinatorics |date=2009 |publisher=Cambridge University Press |isbn=978-0521898065 |page=42}}</ref> More precisely, this sequence corresponds to a [[specifiable combinatorial class]]. The specification of this sequence is <math>\operatorname{Seq}(\mathcal{Z+Z^2})</math>. Indeed, as stated above, the <math>n</math>-th Fibonacci number equals the number of [[Composition (combinatorics)|combinatorial compositions]] (ordered [[Partition (number theory)|partitions]]) of <math>n-1</math> using terms 1 and 2.
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It follows that the [[ordinary generating function]] of the Fibonacci sequence, i.e. <math>\sum_{i=0}^\infty F_iz^i</math>, is the complex function <math>\frac{z}{1-z-z^2}</math>.
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== Other identities ==
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Numerous other identities can be derived using various methods. Some of the most noteworthy are:<ref name="MathWorld">{{MathWorld|urlname=FibonacciNumber |title=Fibonacci Number}}</ref>
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=== Cassini's and Catalan's identities ===
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{{Main|Cassini and Catalan identities}}
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Cassini's identity states that
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<math display=block>F_n^2 - F_{n+1}F_{n-1} = (-1)^{n-1}</math>
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Catalan's identity is a generalization:
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<math display=block>F_n^2 - F_{n+r}F_{n-r} = (-1)^{n-r}F_r^2</math>
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=== d'Ocagne's identity ===
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<math display=block>F_m F_{n+1} - F_{m+1} F_n = (-1)^n F_{m-n}</math>
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<math display=block>F_{2n} = F_{n+1}^2 - F_{n-1}^2 = F_n \left (F_{n+1}+F_{n-1} \right ) = F_nL_n</math>
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where ''L''<sub>''n''</sub> is the ''n'''th [[Lucas number]]. The last is an identity for doubling ''n''; other identities of this type are
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<math display=block>F_{3n} = 2F_n^3 + 3F_n F_{n+1} F_{n-1} = 5F_n^3 + 3 (-1)^n F_n</math>
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by Cassini's identity.
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<math display=block>F_{3n+1} = F_{n+1}^3 + 3 F_{n+1}F_n^2 - F_n^3</math>
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<math display=block>F_{3n+2} = F_{n+1}^3 + 3 F_{n+1}^2F_n + F_n^3</math>
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<math display=block>F_{4n} = 4F_nF_{n+1} \left (F_{n+1}^2 + 2F_n^2 \right ) - 3F_n^2 \left (F_n^2 + 2F_{n+1}^2 \right )</math>
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These can be found experimentally using [[lattice reduction]], and are useful in setting up the [[special number field sieve]] to [[Factorization|factorize]] a Fibonacci number.
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More generally,<ref name="MathWorld" />
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<math display=block>F_{kn+c} = \sum_{i=0}^k {k\choose i} F_{c-i} F_n^i F_{n+1}^{k-i}.</math>
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or alternatively
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<math display=block>F_{kn+c} = \sum_{i=0}^k {k\choose i} F_{c+i} F_n^i F_{n-1}^{k-i}.</math>
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Putting {{math|1=''k'' = 2}} in this formula, one gets again the formulas of the end of above section [[#Matrix form|Matrix form]].
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== Power series ==
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The [[generating function]] of the Fibonacci sequence is the [[power series]]
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<math display=block>s(x)=\sum_{k=0}^{\infty} F_k x^k.</math>
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This series is convergent for <math>|x| < \frac{1}{\varphi},</math> and its sum has a simple closed-form:<ref>{{Citation | last = Glaister | first = P | title = Fibonacci power series | journal = The Mathematical Gazette | year = 1995 | doi = 10.2307/3618079 | volume = 79 | issue = 486| pages = 521–25 | jstor = 3618079 }}</ref>
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<math display=block>s(x)=\frac{x}{1-x-x^2}</math>
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This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:
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<math display=block>\begin{align}
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s(x) &= \sum_{k=0}^{\infty} F_k x^k \\
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&= F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \\
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&= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\
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&= x + x\sum_{k=0}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \\
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&= x + x s(x) + x^2 s(x).
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\end{align}</math>
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Solving the equation
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<math display=block>s(x)=x+xs(x)+x^2s(x)</math>
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for ''s''(''x'') results in the above closed form.
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Setting {{math|1=''x'' = 1/''k''}}, the closed form of the series becomes
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<math display=block>\sum_{n=0}^\infty \frac{F_n}{k^{n}} = \frac{k}{k^{2}-k-1}.</math>
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In particular, if {{mvar|k}} is an integer greater than 1, then this series converges. Further setting {{math|1=''k'' = 10{{sup|''m''}}}} yields
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<math display=block>\sum_{n = 1}^{\infty}{\frac {F_n}{10^{m(n + 1)}}} = \frac {1}{10^{2m} - 10^{m} - 1}</math>
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for all positive integers {{mvar|m}}.
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Some math puzzle-books present as curious the particular value that comes from {{math|1=''m'' = 1}}, which is <math>\frac{s(1/10)}{10}=\frac{1}{89} = .011235\ldots</math><ref>{{Citation | title = Generating functions of Fibonacci-like sequences and decimal expansions of some fractions | last = Köhler | first = Günter | journal = [[The Fibonacci Quarterly]] | volume = 23 | issue = 1 | pages = 29–35 |date=February 1985 | url = http://www.fq.math.ca/Scanned/23-1/kohler.pdf | access-date = December 31, 2011}}</ref> Similarly, {{math|1=''m'' = 2}} gives
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<math display=block>\frac{s(1/100)}{100}=\frac{1}{9899} = .0001010203050813213455\ldots</math>
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== Reciprocal sums ==
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<!--
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{{Citation
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| last =Borwein
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| first =Jonathan M.
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| authorlink =Jonathan Borwein
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| authorlink2=Peter Borwein|first2=Peter B.|last2= Borwein]]
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| title =Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
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| pages =91–101
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| publisher =Wiley
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|date=July 1998
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| url =http://www.wiley.com/WileyCDA/WileyTitle/productCd-047131515X.html
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| isbn = 978-0-471-31515-5 }}
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It credits some formulae to {{Citation | author = Landau, E. | title = Sur la Série des Invers de Nombres de Fibonacci | journal = Bull. Soc. Math. France | volume = 27 | year = 1899 | pages = 298–300}}
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-->
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Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of [[theta function]]s. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as
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<math display=block>\sum_{k=0}^\infty \frac{1}{F_{2k+1}} = \frac{\sqrt{5}}{4}\vartheta_2^2 \left(0, \frac{3-\sqrt 5}{2}\right) ,</math>
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and the sum of squared reciprocal Fibonacci numbers as
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<math display=block>\sum_{k=1}^\infty \frac{1}{F_k^2} = \frac{5}{24} \left(\vartheta_2^4\left(0, \frac{3-\sqrt 5}{2}\right) - \vartheta_4^4\left(0, \frac{3-\sqrt 5}{2}\right) + 1 \right).</math>
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If we add 1 to each Fibonacci number in the first sum, there is also the closed form
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<math display=block>\sum_{k=0}^\infty \frac{1}{1+F_{2k+1}} = \frac{\sqrt{5}}{2},</math>
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and there is a ''nested'' sum of squared Fibonacci numbers giving the reciprocal of the [[golden ratio]],
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<math display=block>\sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sum_{j=1}^k {F_{j}}^2} = \frac{\sqrt{5}-1}{2}.</math>
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No closed formula for the [[reciprocal Fibonacci constant]]
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<math display=block>\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots</math>
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is known, but the number has been proved [[irrational number|irrational]] by [[Richard André-Jeannin]].<ref>{{citation
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| last = André-Jeannin
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| first = Richard
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| title = Irrationalité de la somme des inverses de certaines suites récurrentes
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| journal = [[Comptes Rendus de l'Académie des Sciences, Série I]]
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| volume = 308
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| year = 1989
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| issue = 19
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| pages = 539–41
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|mr=0999451}}</ref>
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The '''Millin series''' gives the identity<ref>{{MathWorld |title=Millin Series |urlname=MillinSeries}}</ref>
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<math display=block>\sum_{n=0}^{\infty} \frac{1}{F_{2^n}} = \frac{7 - \sqrt{5}}{2},</math>
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which follows from the closed form for its partial sums as ''N'' tends to infinity:
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<math display=block>\sum_{n=0}^N \frac{1}{F_{2^n}} = 3 - \frac{F_{2^N-1}}{F_{2^N}}.</math>
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== Primes and divisibility ==
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=== Divisibility properties ===
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Every third number of the sequence is even and more generally, every ''k''th number of the sequence is a multiple of ''F<sub>k</sub>''. Thus the Fibonacci sequence is an example of a [[divisibility sequence]]. In fact, the Fibonacci sequence satisfies the stronger divisibility property<ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | title = My Numbers, My Friends | publisher = Springer-Verlag | year = 2000}}</ref><ref>{{Citation | last1 = Su | first1 = Francis E | others = et al | publisher = HMC | url = http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | contribution = Fibonacci GCD's, please | year = 2000 | title = Mudd Math Fun Facts | access-date = 2007-02-23 | archive-url = https://web.archive.org/web/20091214092739/http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | archive-date = 2009-12-14 | url-status = dead }}</ref>
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<math display=block>\gcd(F_m,F_n) = F_{\gcd(m,n)}.</math>
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Any three consecutive Fibonacci numbers are pairwise [[Coprime integers|coprime]], which means that, for every ''n'',
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:[[greatest common divisor|gcd]](''F''<sub>''n''</sub>, ''F''<sub>''n''+1</sub>) = gcd(''F''<sub>''n''</sub>, ''F''<sub>''n''+2</sub>) = gcd(''F''<sub>''n''+1</sub>, ''F''<sub>''n''+2</sub>) = 1.
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Every prime number ''p'' divides a Fibonacci number that can be determined by the value of ''p'' modulo&nbsp;5. If ''p'' is congruent to 1 or 4 (mod 5), then ''p'' divides ''F''<sub>''p''&nbsp;−&nbsp;1</sub>, and if ''p'' is congruent to 2 or 3 (mod 5), then, ''p'' divides ''F''<sub>''p''&nbsp;+&nbsp;1</sub>. The remaining case is that ''p''&nbsp;=&nbsp;5, and in this case ''p'' divides ''F''<sub>p</sub>.
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<math display=block>\begin{cases} p =5 & \Rightarrow p \mid F_{p}, \\ p \equiv \pm1 \pmod 5 & \Rightarrow p \mid F_{p-1}, \\ p \equiv \pm2 \pmod 5 & \Rightarrow p \mid F_{p+1}.\end{cases}</math>
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These cases can be combined into a single, non-[[piecewise]] formula, using the [[Legendre symbol]]:<ref>{{citation
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| last = Williams | first = H. C.
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| doi = 10.4153/CMB-1982-053-0
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| issue = 3
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| journal = Canadian Mathematical Bulletin
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| mr = 668957
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| pages = 366–70
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| title = A note on the Fibonacci quotient <math>F_{p-\varepsilon}/p</math>
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| volume = 25
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| year = 1982| hdl = 10338.dmlcz/137492
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| hdl-access = free
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}}. Williams calls this property "well known".</ref>
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<math display=block>p \mid F_{p-\left(\frac{5}{p}\right)}.</math>
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=== Primality testing ===
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The above formula can be used as a primality test in the sense that if
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<math display=block>n \mid F_{n-\left(\frac{5}{n}\right)},</math>
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where the Legendre symbol has been replaced by the [[Jacobi symbol]], then this is evidence that ''n'' is a prime, and if it fails to hold, then ''n'' is definitely not a prime. If ''n'' is composite and satisfies the formula, then ''n'' is a ''Fibonacci pseudoprime''. When ''m'' is large{{snd}}say a 500-bit number{{snd}}then we can calculate ''F''<sub>''m''</sub> (mod ''n'') efficiently using the matrix form. Thus
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<math display=block> \begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n.</math>
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Here the matrix power ''A''<sup>''m''</sup> is calculated using [[modular exponentiation]], which can be [[Modular exponentiation#Matrices|adapted to matrices]].<ref>''Prime Numbers'', Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.</ref>
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=== Fibonacci primes ===
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{{Main|Fibonacci prime}}
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A ''Fibonacci prime'' is a Fibonacci number that is [[prime number|prime]]. The first few are:
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: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... {{OEIS2C|id=A005478}}.
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Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.<ref>{{MathWorld |urlname=FibonacciPrime |title=Fibonacci Prime}}</ref>
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''F''<sub>''kn''</sub> is divisible by ''F''<sub>''n''</sub>, so, apart from ''F''<sub>4</sub> = 3, any Fibonacci prime must have a prime index. As there are [[Arbitrarily large|arbitrarily long]] runs of [[composite number]]s, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
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No Fibonacci number greater than ''F''<sub>6</sub> = 8 is one greater or one less than a prime number.<ref>{{Citation | first = Ross | last = Honsberger | title = Mathematical Gems III | journal = AMS Dolciani Mathematical Expositions | year = 1985 | isbn = 978-0-88385-318-4 | page = 133 | issue = 9}}</ref>
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The only nontrivial [[square number|square]] Fibonacci number is 144.<ref>{{cite journal | last = Cohn | first = J. H. E. | doi = 10.1112/jlms/s1-39.1.537 | journal = The Journal of the London Mathematical Society | mr = 163867 | pages = 537–540 | title = On square Fibonacci numbers | volume = 39 | year = 1964}}</ref> Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.<ref>{{Citation | first = Attila | last = Pethő | title = Diophantine properties of linear recursive sequences II | journal = Acta Mathematica Academiae Paedagogicae Nyíregyháziensis | volume = 17 | year = 2001 | pages = 81–96}}</ref> In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.<ref>{{Citation|first1=Y|last1=Bugeaud|first2=M|last2= Mignotte|first3=S|last3=Siksek|title = Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers | journal = Ann. Math.|volume = 2 | year = 2006 | pages = 969–1018 | issue = 163 | bibcode = 2004math......3046B | arxiv = math/0403046| doi = 10.4007/annals.2006.163.969|s2cid=10266596}}</ref>
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1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.<ref>{{Citation|first=Luo|last=Ming|title = On triangular Fibonacci numbers | journal = Fibonacci Quart. | volume = 27 | issue = 2 | year = 1989 | pages = 98–108 | url = http://www.fq.math.ca/Scanned/27-2/ming.pdf }}</ref>
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No Fibonacci number can be a [[perfect number]].<ref name="Luca2000">{{cite journal | first=Florian | last=Luca | title=Perfect Fibonacci and Lucas numbers | journal=Rendiconti del Circolo Matematico di Palermo | year=2000 | volume=49 | issue=2 | pages=313–18 | doi=10.1007/BF02904236 | mr=1765401 | s2cid=121789033 | issn=1973-4409 }}</ref> More generally, no Fibonacci number other than 1 can be [[multiply perfect number|multiply perfect]],<ref name="BGLLHT2011">{{cite journal | first1=Kevin A. | last1=Broughan | first2=Marcos J. | last2=González | first3=Ryan H. | last3=Lewis | first4=Florian | last4=Luca | first5=V. Janitzio | last5=Mejía Huguet | first6=Alain | last6=Togbé | title=There are no multiply-perfect Fibonacci numbers | journal=Integers | year=2011 | volume=11a | page=A7 | url=http://math.colgate.edu/~integers/vol11a.html | mr=2988067 }}</ref> and no ratio of two Fibonacci numbers can be perfect.<ref name="LucaMH2010">{{cite journal | first1=Florian | last1=Luca | first2= V. Janitzio | last2=Mejía Huguet | title=On Perfect numbers which are ratios of two Fibonacci numbers | journal=Annales Mathematicae at Informaticae | year=2010 | volume=37 | pages=107–24 | url=http://ami.ektf.hu/index.php?vol=37 | mr=2753031 | issn=1787-6117 }}</ref>
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=== Prime divisors ===
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With the exceptions of 1, 8 and 144 (''F''<sub>1</sub> = ''F''<sub>2</sub>, ''F''<sub>6</sub> and ''F''<sub>12</sub>) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number ([[Carmichael's theorem]]).<ref>{{Citation | first = Ron | last = Knott | url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html | title = The Fibonacci numbers | publisher = Surrey | place = UK}}</ref> As a result, 8 and 144 (''F''<sub>6</sub> and ''F''<sub>12</sub>) are the only Fibonacci numbers that are the product of other Fibonacci numbers {{OEIS2C|id=A235383}}.
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The divisibility of Fibonacci numbers by a prime ''p'' is related to the [[Legendre symbol]] <math>\left(\tfrac{p}{5}\right)</math> which is evaluated as follows:
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<math display=block>\left(\frac{p}{5}\right) = \begin{cases} 0 & \text{if } p = 5\\ 1 & \text{if } p \equiv \pm 1 \pmod 5\\ -1 & \text{if } p \equiv \pm 2 \pmod 5.\end{cases}</math>
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If ''p'' is a prime number then
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<math display=block> F_p \equiv \left(\frac{p}{5}\right) \pmod p \quad \text{and}\quad F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p.</math><ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | year = 1996 | title = The New Book of Prime Number Records | place = New York | publisher = Springer | isbn = 978-0-387-94457-9 | page = 64}}</ref>{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.25–28 | pp = 73–74}}
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For example,
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<math display=block>\begin{align}
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(\tfrac{2}{5}) &= -1, &F_3 &= 2, &F_2&=1, \\
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(\tfrac{3}{5}) &= -1, &F_4 &= 3,&F_3&=2, \\
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(\tfrac{5}{5}) &= 0, &F_5 &= 5, \\
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(\tfrac{7}{5}) &= -1, &F_8 &= 21,&F_7&=13, \\
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(\tfrac{11}{5})& = +1, &F_{10}& = 55, &F_{11}&=89.
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\end{align}</math>
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It is not known whether there exists a prime ''p'' such that
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<math display=block>F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod{p^2}.</math>
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Such primes (if there are any) would be called [[Wall–Sun–Sun prime]]s.
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Also, if ''p'' ≠ 5 is an odd prime number then:{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.28 | pp = 73–74}}
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<math display=block>5F^2_{\frac{p \pm 1}{2}} \equiv \begin{cases}
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\tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\pm 5 \right ) \pmod p & \text{if } p \equiv 1 \pmod 4\\
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\tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\mp 3 \right ) \pmod p & \text{if } p \equiv 3 \pmod 4.
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\end{cases}</math>
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'''Example 1.''' ''p'' = 7, in this case ''p'' ≡ 3 (mod 4) and we have:
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<math display=block>(\tfrac{7}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{7}{5})+3 \right ) =-1, \quad \tfrac{1}{2} \left (5(\tfrac{7}{5})-3 \right )=-4.</math>
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<math display=block>F_3=2 \text{ and } F_4=3.</math>
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<math display=block>5F_3^2=20\equiv -1 \pmod {7}\;\;\text{ and }\;\;5F_4^2=45\equiv -4 \pmod {7}</math>
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'''Example 2.''' ''p'' = 11, in this case ''p'' ≡ 3 (mod 4) and we have:
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<math display=block>(\tfrac{11}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{11}{5})+3 \right )=4, \quad \tfrac{1}{2} \left (5(\tfrac{11}{5})- 3 \right )=1.</math>
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<math display=block>F_5=5 \text{ and } F_6=8.</math>
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<math display=block>5F_5^2=125\equiv 4 \pmod {11} \;\;\text{ and }\;\;5F_6^2=320\equiv 1 \pmod {11}</math>
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'''Example 3.''' ''p'' = 13, in this case ''p'' ≡ 1 (mod 4) and we have:
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<math display=block>(\tfrac{13}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{13}{5})-5 \right ) =-5, \quad \tfrac{1}{2}\left (5(\tfrac{13}{5})+ 5 \right )=0.</math>
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<math display=block>F_6=8 \text{ and } F_7=13.</math>
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<math display=block>5F_6^2=320\equiv -5 \pmod {13} \;\;\text{ and }\;\;5F_7^2=845\equiv 0 \pmod {13}</math>
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'''Example 4.''' ''p'' = 29, in this case ''p'' ≡ 1 (mod 4) and we have:
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<math display=block>(\tfrac{29}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{29}{5})-5 \right )=0, \quad \tfrac{1}{2}\left (5(\tfrac{29}{5})+5 \right )=5.</math>
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<math display=block>F_{14}=377 \text{ and } F_{15}=610.</math>
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<math display=block>5F_{14}^2=710645\equiv 0 \pmod {29} \;\;\text{ and }\;\;5F_{15}^2=1860500\equiv 5 \pmod {29}</math>
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For odd ''n'', all odd prime divisors of ''F''<sub>''n''</sub> are congruent to 1 modulo 4, implying that all odd divisors of ''F''<sub>''n''</sub> (as the products of odd prime divisors) are congruent to 1 modulo 4.{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.27 | p = 73}}
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For example,
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<math display=block>F_1 = 1, F_3 = 2, F_5 = 5, F_7 = 13, F_9 = 34 = 2 \cdot 17, F_{11} = 89, F_{13} = 233, F_{15} = 610 = 2 \cdot 5 \cdot 61.</math>
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All known factors of Fibonacci numbers ''F''(''i'') for all ''i'' < 50000 are collected at the relevant repositories.<ref>{{Citation | url = http://mersennus.net/fibonacci/ | title = Fibonacci and Lucas factorizations | publisher = Mersennus}} collects all known factors of ''F''(''i'') with ''i'' < 10000.</ref><ref>{{Citation | url =http://fibonacci.redgolpe.com/ | title = Factors of Fibonacci and Lucas numbers | publisher = Red golpe}} collects all known factors of ''F''(''i'') with 10000 < ''i'' < 50000.</ref>
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=== Periodicity modulo ''n'' ===
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{{Main|Pisano period}}
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If the members of the Fibonacci sequence are taken mod&nbsp;''n'', the resulting sequence is [[periodic sequence|periodic]] with period at most&nbsp;''6n''.<ref>{{Citation | title = Problems and Solutions: Solutions: E3410 | last1 = Freyd | first1 = Peter | last2 = Brown | first2 = Kevin S. | journal = The American Mathematical Monthly | volume = 99 | issue = 3 | pages = 278–79 |date= 1993 | doi=10.2307/2325076| jstor = 2325076 }}</ref> The lengths of the periods for various ''n'' form the so-called [[Pisano period]]s {{OEIS2C|id=A001175}}. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the [[multiplicative order]] of a [[modular arithmetic|modular integer]] or of an element in a [[finite field]]. However, for any particular ''n'', the Pisano period may be found as an instance of [[cycle detection]].
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== Magnitude ==
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Since ''F<sub>n</sub>'' is [[Asymptotic analysis|asymptotic]] to <math>\varphi^n/\sqrt5</math>, the number of digits in ''F''<sub>''n''</sub> is asymptotic to <math>n\log_{10}\varphi\approx 0.2090\, n</math>. As a consequence, for every integer ''d'' > 1 there are either 4 or 5 Fibonacci numbers with ''d'' decimal digits.
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More generally, in the base ''b'' representation, the number of digits in ''F''<sub>''n''</sub> is asymptotic to <math>n\log_b\varphi.</math>
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== Generalizations ==
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{{Main|Generalizations of Fibonacci numbers}}
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The Fibonacci sequence is one of the simplest and earliest known sequences defined by a [[recurrence relation]], and specifically by a [[linear difference equation]]. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.
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Some specific examples that are close, in some sense, from Fibonacci sequence include:
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* Generalizing the index to negative integers to produce the [[#Negafibonacci|negafibonacci]] numbers.
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* Generalizing the index to real numbers using a modification of Binet's formula.<ref name="MathWorld" />
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* Starting with other integers. [[Lucas number]]s have ''L''<sub>1</sub> = 1, ''L''<sub>2</sub> = 3, and ''L<sub>n</sub>'' = ''L''<sub>''n''−1</sub> + ''L''<sub>''n''−2</sub>. [[Primefree sequence]]s use the Fibonacci recursion with other starting points to generate sequences in which all numbers are [[composite number|composite]].
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* Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The [[Pell number]]s have ''P<sub>n</sub>'' = 2''P''<sub>''n'' − 1</sub> + ''P''<sub>''n'' − 2</sub>. If the coefficient of the preceding value is assigned a variable value ''x'', the result is the sequence of [[Fibonacci polynomials]].
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* Not adding the immediately preceding numbers. The [[Padovan sequence]] and [[Perrin number]]s have ''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3).
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* Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as ''n-Step Fibonacci numbers''.<ref>{{MathWorld|urlname=Fibonaccin-StepNumber |title=Fibonacci ''n''-Step Number}}</ref>
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== Applications ==
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[[File:PascalTriangleFibanacci.svg|thumb|right|360px|The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of [[Pascal's triangle]].]]
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The Fibonacci numbers occur in the sums of "shallow" diagonals in [[Pascal's triangle]] (see [[binomial coefficient]]):{{Sfn | Lucas | 1891 | p = 7}}
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=== Mathematics ===
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<math display=block>F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}.</math>
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[[File:Fibonacci_climbing_stairs.svg|thumb|right|Use of the Fibonacci sequence to count {{nowrap|{1, 2}-restricted}} compositions]]
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These numbers also give the solution to certain enumerative problems,<ref>{{cite book|last=Stanley|first=Richard|title=Enumerative Combinatorics I (2nd ed.)|year=2011|publisher=Cambridge Univ. Press|isbn=978-1-107-60262-5|page=121, Ex 1.35}}</ref> the most common of which is that of counting the number of ways of writing a given number {{math|''n''}} as an ordered sum of 1s and 2s (called [[composition_(combinatorics)#Number_of_compositions|compositions]]); there are {{math|''F''<sub>''n''+1</sub>}} ways to do this. For example, there are {{math|1=''F''<sub>5+1</sub> = ''F''<sub>6</sub> = 8}} ways one can climb a staircase of 5 steps, taking one or two steps at a time:
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:{|
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| {{math|5}}
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| {{math|1== 1+1+1+1+1}}
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| {{math|1== 2+1+1+1}}
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| {{math|1== 1+2+1+1}}
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| {{math|1== 1+1+2+1}}
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| {{math|1== 2+2+1}}
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|-
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|
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| {{math|1== 1+1+1+2}}
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| {{math|1== 2+1+2}}
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| {{math|1== 1+2+2}}
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|}
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The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied [[recursion|recursively]] until a single step, of which there is only one way to climb.
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The Fibonacci numbers can be found in different ways among the set of [[binary numeral system|binary]] [[String (computer science)|strings]], or equivalently, among the [[subset]]s of a given set.
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* The number of binary strings of length {{math|''n''}} without consecutive {{math|1}}s is the Fibonacci number {{math|''F''<sub>''n''+2</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>6</sub> = 8}} without consecutive {{math|1}}s&nbsp;– they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Equivalently, {{math|''F''<sub>''n''+2</sub>}} is the number of subsets {{math|''S''}}{{math|}} of {{math|{{mset|1, ..., ''n''}}}} without consecutive integers, that is, those {{math|''S''}} for which {{math|{{mset|''i'', ''i'' + 1}} ⊈ ''S''}} for every {{math|''i''}}.
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* The number of binary strings of length {{math|''n''}} without an odd number of consecutive {{math|1}}s is the Fibonacci number {{math|''F''<sub>''n+1''</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>5</sub> = 5}} without an odd number of consecutive {{math|1}}s&nbsp;– they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets {{math|''S''}} of {{math|{{mset|1, ..., ''n''}}}} without an odd number of consecutive integers is {{math|''F''<sub>''n''+1</sub>}}.
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* The number of binary strings of length {{math|''n''}} without an even number of consecutive {{math|0}}s or {{math|1}}s is {{math|2''F''<sub>''n''</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=2''F''<sub>4</sub> = 6}} without an even number of consecutive {{math|0}}s or {{math|1}}s&nbsp;– they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
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*[[Yuri Matiyasevich]] was able to show that the Fibonacci numbers can be defined by a [[Diophantine equation]], which led to [[Matiyasevich's theorem|his solving]] [[Hilbert's tenth problem]].<ref>{{citation|title=Review of Yuri V. Matiyasevich, ''Hibert's Tenth Problem''|journal=Modern Logic|first=Valentina|last=Harizanov|volume=5|issue=3|year=1995|pages=345–55|url=http://projecteuclid.org/euclid.rml/1204900767}}.</ref>
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*The Fibonacci numbers are also an example of a [[complete sequence]]. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
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*Moreover, every positive integer can be written in a unique way as the sum of ''one or more'' distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as [[Zeckendorf's theorem]], and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its [[Fibonacci coding]].
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*Starting with 5, every second Fibonacci number is the length of the hypotenuse of a [[right triangle]] with integer sides, or in other words, the largest number in a [[Pythagorean triple]], obtained from the formula <math display=block>(F_nF_{n+3})^2 + (2F_{n+1}F_{n+2})^2 = F_{2n+3}^2.</math> The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... The middle side of each of these triangles is the sum of the three sides of the preceding triangle.<ref>{{citation
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| last = Pagni | first = David
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| date = September 2001
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| issue = 4
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| journal = Mathematics in School
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| jstor = 30215477
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| pages = 39–40
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| title = Fibonacci Meets Pythagoras
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| volume = 30}}</ref>
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*The [[Fibonacci cube]] is an [[undirected graph]] with a Fibonacci number of nodes that has been proposed as a [[network topology]] for [[parallel computing]].
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*Fibonacci numbers appear in the [[ring lemma]], used to prove connections between the [[circle packing theorem]] and [[conformal map]]s.<ref>{{citation|last=Stephenson|first=Kenneth|isbn=978-0-521-82356-2|mr=2131318|publisher=Cambridge University Press|title=Introduction to Circle Packing: The Theory of Discrete Analytic Functions|title-link=Introduction to Circle Packing|year=2005}}; see especially Lemma 8.2 (Ring Lemma), [https://books.google.com/books?id=38PxEmKKhysC&pg=PA73 pp. 73–74], and Appendix B, The Ring Lemma, pp. 318–321.</ref>
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=== Computer science ===
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[[File:Fibonacci Tree 6.svg|thumb|right|360px|Fibonacci tree of height 6. [[AVL tree#Balance factor|Balance factor]]s green; heights red.<br />The keys in the left spine are Fibonacci numbers.]]
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*The Fibonacci numbers are important in [[Analysis of algorithms|computational run-time analysis]] of [[Euclidean algorithm|Euclid's algorithm]] to determine the [[greatest common divisor]] of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.<ref>{{Citation| first= Donald E |last= Knuth| author-link= Donald Knuth | year =1997|title=The Art of Computer Programming | volume = 1: Fundamental Algorithms|edition= 3rd | publisher = Addison–Wesley |isbn=978-0-201-89683-1 | page = 343}}</ref>
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*Fibonacci numbers are used in a polyphase version of the [[merge sort]] algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers&nbsp;– by dividing the list so that the two parts have lengths in the approximate proportion ''φ''. A tape-drive implementation of the [[polyphase merge sort]] was described in ''[[The Art of Computer Programming]]''.
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*{{anchor|Fibonacci Tree}}A Fibonacci tree is a [[binary tree]] whose child trees (recursively) differ in [[Tree height|height]] by exactly 1. So it is an [[AVL tree]], and one with the fewest nodes for a given height — the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.<ref>{{cite journal|last1=Adelson-Velsky|first1=Georgy|last2=Landis|first2=Evgenii|year=1962|title=An algorithm for the organization of information|journal=[[Proceedings of the USSR Academy of Sciences]]|volume=146|pages=263–266|language=ru}} [https://zhjwpku.com/assets/pdf/AED2-10-avl-paper.pdf English translation] by Myron J. Ricci in ''Soviet Mathematics - Doklady'', 3:1259–1263, 1962.</ref>
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*Fibonacci numbers are used by some [[pseudorandom number generators]].<!-- Knuth vol. 2 -->
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*Fibonacci numbers arise in the analysis of the [[Fibonacci heap]] data structure.
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*A one-dimensional optimization method, called the [[Fibonacci search technique]], uses Fibonacci numbers.<ref>{{Citation| first1 = M | last1 = Avriel | first2 = DJ | last2 = Wilde | title= Optimality of the Symmetric Fibonacci Search Technique |journal=Fibonacci Quarterly|year=1966 |issue=3 |pages= 265–69}}</ref>
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*The Fibonacci number series is used for optional [[lossy compression]] in the [[Interchange File Format|IFF]] [[8SVX]] audio file format used on [[Amiga]] computers. The number series [[companding|compands]] the original audio wave similar to logarithmic methods such as [[μ-law]].<ref>{{Citation | title = Amiga ROM Kernel Reference Manual | publisher = Addison–Wesley | year = 1991}}</ref><ref>{{Citation | url = http://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression | contribution = IFF | title = Multimedia Wiki}}</ref>
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*They are also used in [[planning poker]], which is a step in estimating in software development projects that use the [[Scrum (software development)|Scrum]] methodology.
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=== Nature ===
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{{Further |Patterns in nature}}
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{{see also|Golden ratio#Nature}}
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[[File:FibonacciChamomile.PNG|thumb|[[Yellow chamomile]] head showing the arrangement in 21 (blue) and 13 (aqua) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.]]
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Fibonacci sequences appear in biological settings,<ref>{{Citation |first1=S |last1=Douady |first2=Y |last2=Couder |title=Phyllotaxis as a Dynamical Self Organizing Process |journal=Journal of Theoretical Biology |year=1996 |issue=3 |pages=255–74 |url=http://www.math.ntnu.no/~jarlet/Douady96.pdf |doi=10.1006/jtbi.1996.0026 |volume=178 |url-status=dead |archive-url=https://web.archive.org/web/20060526054108/http://www.math.ntnu.no/~jarlet/Douady96.pdf |archive-date=2006-05-26 }}</ref> such as branching in trees, [[Phyllotaxis|arrangement of leaves on a stem]], the fruitlets of a [[pineapple]],<ref>{{Citation | first1=Judy |last1=Jones | first2=William | last2=Wilson |title=An Incomplete Education |publisher=Ballantine Books |year=2006 |isbn=978-0-7394-7582-9 |page=544 |chapter=Science}}</ref> the flowering of [[artichoke]], an uncurling fern and the arrangement of a [[pine cone]],<ref>{{Citation| first=A | last=Brousseau |title=Fibonacci Statistics in Conifers | journal=[[Fibonacci Quarterly]] |year=1969 |issue=7 |pages=525–32}}</ref> and the family tree of honeybees.<ref>{{cite web|url = http://www.cs4fn.org/maths/bee-davinci.php |work = Maths | publisher = Computer Science For Fun: CS4FN |title = Marks for the da Vinci Code: B–}}</ref><ref>{{Citation|first1=T.C.|last1=Scott|first2=P.|last2=Marketos| url = http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf | title = On the Origin of the Fibonacci Sequence | publisher = [[MacTutor History of Mathematics archive]], University of St Andrews| date = March 2014}}</ref> [[Kepler]] pointed out the presence of the Fibonacci sequence in nature, using it to explain the ([[golden ratio]]-related) pentagonal form of some flowers.{{sfn|Livio|2003|p=110}} Field [[Leucanthemum vulgare|daisies]] most often have petals in counts of Fibonacci numbers.{{sfn|Livio|2003|pp=112–13}} In 1754, [[Charles Bonnet]] discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.<ref>{{cite web|url=http://www.amnh.org:80/learn-teach/young-naturalist-awards/winning-essays/2011/the-secret-of-the-fibonacci-sequence-in-trees|title=The Secret of the Fibonacci Sequence in Trees|year=2011|website=[[American Museum of Natural History]]|archive-url=https://web.archive.org/web/20130504125059/http://www.amnh.org/learn-teach/young-naturalist-awards/winning-essays/2011/the-secret-of-the-fibonacci-sequence-in-trees|archive-date=4 May 2013|access-date=4 February 2019|url-status=live}}</ref>
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[[Przemysław Prusinkiewicz]] advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on [[free group]]s, specifically as certain [[L-system|Lindenmayer grammars]].<ref>{{Citation|first1 = Przemyslaw |last1 = Prusinkiewicz | first2 = James | last2 = Hanan| title = Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics) |publisher= [[Springer Science+Business Media|Springer-Verlag]] |year=1989 |isbn=978-0-387-97092-9}}</ref>
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[[File:SunflowerModel.svg|thumb|Illustration of Vogel's model for {{math|''n'' {{=}} 1 ... 500}}]]
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A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by {{ill|Helmut Vogel|de|Helmut Vogel (Physiker)}} in 1979.<ref>{{Citation | last =Vogel | first =Helmut | title =A better way to construct the sunflower head | journal = Mathematical Biosciences | issue =3–4 | pages = 179–89 | year = 1979 | doi = 10.1016/0025-5564(79)90080-4 | volume = 44}}</ref> This has the form
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<math display=block>\theta = \frac{2\pi}{\varphi^2} n,\ r = c \sqrt{n}</math>
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where {{math|''n''}} is the index number of the floret and {{math|''c''}} is a constant scaling factor; the florets thus lie on [[Fermat's spiral]]. The divergence angle, approximately 137.51°, is the [[golden angle]], dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form {{math|''F''(''j''):''F''(''j'' + 1)}}, the nearest neighbors of floret number {{math|''n''}} are those at {{math|''n'' ± ''F''(''j'')}} for some index {{math|''j''}}, which depends on {{math|''r''}}, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,{{sfn|Livio|2003|p=112}} typically counted by the outermost range of radii.<ref>{{Citation | last1 = Prusinkiewicz | first1 = Przemyslaw | author1-link = Przemyslaw Prusinkiewicz | author2-link = Aristid Lindenmayer | last2 = Lindenmayer | first2 = Aristid | title = The Algorithmic Beauty of Plants | publisher = Springer-Verlag | year = 1990 | pages = [https://archive.org/details/algorithmicbeaut0000prus/page/101 101–107] | chapter = 4 | chapter-url = http://algorithmicbotany.org/papers/#webdocs | isbn = 978-0-387-97297-8 | url = https://archive.org/details/algorithmicbeaut0000prus/page/101 }}</ref>
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Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:
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* If an egg is laid by an unmated female, it hatches a male or [[Drone (bee)|drone bee]].
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* If, however, an egg was fertilized by a male, it hatches a female.
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Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, {{math|''F''<sub>''n''</sub>}}, is the number of female ancestors, which is {{math|''F''<sub>''n''−1</sub>}}, plus the number of male ancestors, which is {{math|''F''<sub>''n''−2</sub>}}.<ref>{{Citation | url = http://www.fq.math.ca/Scanned/1-1/basin.pdf | title = The Fibonacci sequence as it appears in nature | journal = The Fibonacci Quarterly | volume = 1 | number = 1 | pages = 53–56 | year = 1963}}</ref> This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
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[[File:X chromosome ancestral line Fibonacci sequence.svg|thumb|360px|The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".<ref name="xcs"/>)]]
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It has been noticed that the number of possible ancestors on the human [[X chromosome]] inheritance line at a given ancestral generation also follows the Fibonacci sequence.<ref name="xcs">{{Cite journal|last=Hutchison|first=Luke|date=September 2004|title=Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships|url=http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|journal=Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)|access-date=2016-09-03}}</ref> A male individual has an X chromosome, which he received from his mother, and a [[Y chromosome]], which he received from his father. The male counts as the "origin" of his own X chromosome (<math>F_1=1</math>), and at his parents' generation, his X chromosome came from a single parent (<math>F_2=1</math>). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome (<math>F_3=2</math>). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome (<math>F_4=3</math>). Five great-great-grandparents contributed to the male descendant's X chromosome (<math>F_5=5</math>), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a [[Founder effect|population founder]] appears on all lines of the genealogy.)
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The pathways of [[tubulin]]s on intracellular [[microtubule]]s arrange in patterns of 3, 5, 8 and 13.<ref>{{cite journal|title=Consciousness in the universe: A review of the 'Orch OR' theory|last1=Hameroff|first1=Stuart|last2=Penrose|first2=Roger|date=March 2014|journal=Physics of Life Reviews|publisher=Elsevier|volume=11|issue=1|pages=39–78|doi=10.1016/j.plrev.2013.08.002|pmid=24070914|bibcode=2014PhLRv..11...39H|doi-access=free}}</ref>
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===Other===
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*In [[optics]], when a beam of light shines at an angle through two stacked transparent plates of different materials of different [[refractive index]]es, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have {{mvar|k}} reflections, for {{math|''k'' > 1}}, is the <math>k</math>th Fibonacci number. (However, when {{math|1=''k'' = 1}}, there are three reflection paths, not two, one for each of the three surfaces.){{sfn|Livio|2003|pp=98–99}}
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* [[Fibonacci retracement]] levels are widely used in [[technical analysis]] for financial market trading.
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*Since the [[conversion of units|conversion]] factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a [[radix]] 2 number [[processor register|register]] in [[golden ratio base]] ''φ'' being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<ref>{{Citation | url = http://www.encyclopediaofmath.org/index.php/Zeckendorf_representation | contribution = Zeckendorf representation | title = Encyclopedia of Math}}</ref>
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*Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.<ref name="Brasch et al. 2012">{{Citation| first1 =T. von | last1 = Brasch | first2 = J. | last2 = Byström | first3 = L.P. | last3 = Lystad| title= Optimal Control and the Fibonacci Sequence |journal = Journal of Optimization Theory and Applications |year=2012 |issue=3 |pages= 857–78 |doi = 10.1007/s10957-012-0061-2
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|volume=154 | hdl = 11250/180781 | s2cid = 8550726 | hdl-access = free }}</ref> In particular, it is shown how a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
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*[[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}
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*[[Joseph Schillinger]] (1895–1943) developed a [[Schillinger System|system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.
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== See also ==
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{{Div col}}
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*[[Elliott wave principle]]
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*[[Embree–Trefethen constant]]
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*[[The Fibonacci Association]]
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*[[Fibonacci numbers in popular culture]]
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*[[Fibonacci word]]
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*[[Strong Law of Small Numbers]]
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*[[Verner Emil Hoggatt Jr.]]
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*[[Wythoff array]]
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*[[Fibonacci retracement]]
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{{Div col end}}
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== References ==
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'''Footnotes'''
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{{notelist}}
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'''Citations'''
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{{Reflist|30em}}
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===Works cited===
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* {{Citation | title= Strange Curves, Counting Rabbits, and Other Mathematical Explorations | first= Keith M | last = Ball |publisher= [[Princeton University Press]]| place= Princeton, NJ | year= 2003 | chapter= 8: Fibonacci's Rabbits Revisited |isbn= 978-0-691-11321-0}}.
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* {{Citation |title= The Art of Proof: Basic Training for Deeper Mathematics |first1= Matthias |last1= Beck |first2 = Ross |last2=Geoghegan |publisher=Springer |place=New York |year= 2010 |isbn=978-1-4419-7022-0}}.
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* {{Citation |title=A Walk Through Combinatorics |edition= 3rd |first= Miklós |last= Bóna |author-link=Miklós Bóna |publisher= World Scientific | place=New Jersey |year= 2011 |isbn= 978-981-4335-23-2}}.
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** {{Citation |title=A Walk Through Combinatorics |edition= 4th Revised |first= Miklós |last= Bóna |publisher= World Scientific | place=New Jersey |year= 2016 |isbn= 978-981-3148-84-0}}.
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* {{Citation | first = Franz | last = Lemmermeyer | year = 2000 | title = Reciprocity Laws: From Euler to Eisenstein | series = Springer Monographs in Mathematics | place = New York | publisher = Springer | isbn = 978-3-540-66957-9}}.
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* {{Cite book | last = Livio | first = Mario | author-link = Mario Livio | title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number | url = https://books.google.com/books?id=bUARfgWRH14C | orig-year = 2002 | edition = First trade paperback | year = 2003 | publisher = [[Random House|Broadway Books]] | location = New York City | isbn = 0-7679-0816-3 }}
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* {{Citation |title=Théorie des nombres |first= Édouard |last= Lucas |publisher= Gauthier-Villars|year= 1891 | volume = 1 | language = fr | place = Paris | url = https://archive.org/details/thoriedesnombr01lucauoft | id = {{Google books|_hsPAAAAIAAJ|Théorie des nombres|plainurl=yes}} }}.
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* {{Citation | first = Leonardo | last = Pisano | title = Fibonacci's Liber Abaci: A Translation into Modern English of the Book of Calculation | others = Sigler, Laurence E, trans |series=Sources and Studies in the History of Mathematics and Physical Sciences | publisher=Springer | year=2002 | isbn=978-0-387-95419-6}}
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== External links ==
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{{Wikiquote}}
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{{Wikibooks|Fibonacci number program}}
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* [http://www.mathpages.com/home/kmath078/kmath078.htm Periods of Fibonacci Sequences Mod m] at MathPages
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* [http://www.physorg.com/news97227410.html Scientists find clues to the formation of Fibonacci spirals in nature]
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* {{In Our Time|Fibonacci Sequence|b008ct2j|Fibonacci_Sequence}}
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* {{springer|title=Fibonacci numbers|id=p/f040020}}
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* {{OEIS el|sequencenumber=A000045|name=Fibonacci numbers|formalname=Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1}}
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{{Classes of natural numbers}}
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{{Metallic ratios}}
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{{Series (mathematics)}}
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{{Fibonacci}}
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{{Authority control}}
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{{Interwiki extra|qid=Q23835349}}
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[[Category:Fibonacci numbers| ]]
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[[Category:Articles containing proofs]]
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bottom listen other two your
world should old heard parent
eyes yell arctic suit finish
board live city planet affect
bought scare would return
they held broken there visit
could most really today sign
once young what wear choose
move women unique water lose
find earth enough volume
color door work quite pretty
sure right you field turtle
radio future word from minute
peanut doctor reason laugh
island was easily among
picnic want walk ocean caught
again any night vacuum pencil
else during genius arrive
watch height give enjoy enemy
hoping come some later great
idea since have group catch
safety few raise write police
who middle office been often
tired prize origin both kind
were edge excuse mother
toward except slowly friend
only become cousin people
advice talk answer their
sudden charge learn heavy
dollar belief off decide said
are tried across until many
piece snow threw where know
bottom listen other two your
world should old heard parent
eyes yell arctic suit finish
board live city planet affect
bought scare would return
they held broken there visit
could most really today sign
once young what wear choose
move women unique water lose
find earth enough volume
color door work quite pretty
sure right you field turtle
radio future word from minute
peanut doctor reason laugh
island was easily among
picnic want walk ocean caught
again any night vacuum pencil
else during genius arrive
watch height give enjoy enemy
hoping come some later great
idea since have group catch
safety few raise write police
who middle office been often
tired prize origin both kind
were edge excuse mother
toward except slowly friend
only become cousin people
advice talk answer their
sudden charge learn heavy
dollar belief off decide said
are tried across until many
piece snow threw where know
bottom listen other two your
world should old heard parent
eyes yell arctic suit finish
board live city planet affect
bought scare would return
they held broken there visit
could most really today sign
once young what wear choose
move women unique water lose
find earth enough volume
color door work quite pretty
sure right you field turtle
radio future word from minute
peanut doctor reason laugh
island was easily among
picnic want walk ocean caught
again any night vacuum pencil
else during genius arrive
watch height give enjoy enemy
hoping come some later great
idea since have group catch
safety few raise write police
who middle office been often
tired prize origin both kind
were edge excuse mother
toward except slowly friend
only become cousin people
advice talk answer their
sudden charge learn heavy
dollar belief off decide said
are tried across until many
piece snow threw where know
bottom listen other two your
world should old heard parent
eyes yell arctic suit finish
board live city planet affect
bought scare would return
they held broken there visit
could most really today sign
once young what wear choose
move women unique water lose
find earth enough volume
color door work quite pretty
sure right you field turtle
radio future word from minute
peanut doctor reason laugh
island was easily among
picnic want walk ocean caught
again any night vacuum pencil
else during genius arrive
watch height give enjoy enemy
hoping come some later great
idea since have group catch
safety few raise write police
who middle office been often
tired prize origin both kind
were edge excuse mother
toward except slowly friend
only become cousin people
advice talk answer their
sudden charge learn heavy
dollar belief off decide said
are tried across until many
piece snow threw where know
bottom listen other two your
world should old heard parent
eyes yell arctic suit finish
board live city planet affect
bought scare would return
they held broken there visit
could most really today sign
once young what wear choose
move women unique water lose
find earth enough volume
color door work quite pretty
sure right you field turtle
radio future word from minute
peanut doctor reason laugh
island was easily among
picnic want walk ocean caught
again any night vacuum pencil
else during genius arrive
watch height give enjoy enemy
hoping come some later great
idea since have group catch
safety few raise write police

Revision as of 00:45, 11 May 2021

who middle office been often tired prize origin both kind were edge excuse, mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your the world should old heard parent eyes yell arctic suit finish board live city-planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from the minute peanut doctor reason laugh the island was easily among picnic want walk ocean-caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since having group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse, mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your the world should old heard parent eyes yell arctic suit finish board live city-planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from the minute peanut doctor reason laugh the island was easily among picnic want walk ocean-caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since having group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse, mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your the world should old heard parent eyes yell arctic suit finish board live city-planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from the minute peanut doctor reason laugh the island was easily among picnic want walk ocean-caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since having group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse, mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your the world should old heard parent eyes yell arctic suit finish board live city-planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from the minute peanut doctor reason laugh the island was easily among picnic want walk ocean-caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since having group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse, mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your the world should old heard parent eyes yell arctic suit finish board live city-planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from the minute peanut doctor reason laugh the island was easily among picnic want walk ocean-caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since having group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police who middle office been often tired prize origin both kind were edge excuse mother toward except slowly friend only become cousin people advice talk answer their sudden charge learn heavy dollar belief off decide said are tried across until many piece snow threw where know bottom listen other two your world should old heard parent eyes yell arctic suit finish board live city planet affect bought scare would return they held broken there visit could most really today sign once young what wear choose move women unique water lose find earth enough volume color door work quite pretty sure right you field turtle radio future word from minute peanut doctor reason laugh island was easily among picnic want walk ocean caught again any night vacuum pencil else during genius arrive watch height give enjoy enemy hoping come some later great idea since have group catch safety few raise write police

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