Talk:Fibonacci sequence

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Requested move 1 March 2023

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved to Fibonacci sequence. In this discussion, both the current title and the proposed title proved to be controversial. The most hotly debated question was in the interpretation of WP:PLURAL – should this article be pluralized as a topic that is naturally the set or family of [numbers], or should it remain singular under the standard guidance? Both interpretations were supported by various participants, and neither side appeared to convince the other. For other angles of argument, the singular title was shown to be more WP:CONSISTENT with other articles about classes of numbers, while the plural title was shown to be a more common name than the singular. This prevented a consensus from forming as to whether "Fibonacci numbers" was preferable to "Fibonacci number"... but those two titles were not the only titles proposed in the discussion.

The title "Fibonacci sequence" was proposed early in the discussion, and attracted substantial support. Arguments in favor of "Fibonacci sequence" included its wide WP:RECOGNIZABILITY and its avoidance of problems with the other names (the "sequence" title means consistency with number articles becomes a non-issue, and sources use it with a similar frequency to the plural title). "Fibonacci sequence" was also identified as an acceptable compromise title by several users whose first preference was something different. Support for "Fibonacci sequence" was not perfectly unanimous – one user argued that the article is more WP:PRECISEly about the numbers rather than the sequence, and another argued that "Fibonacci sequence" was not sufficiently the WP:COMMONNAME – but the arguments against "Fibonacci sequence" were contested by other editors, and ultimately did not sway the overall tide of support. Consequently, I find a consensus for moving the article to Fibonacci sequence. (non-admin closure) ModernDayTrilobite (talk • contribs) 15:44, 10 March 2023 (UTC)

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Fibonacci number → Fibonacci numbers – The plural makes more sense, because it is a sequence of numbers, not just one singular Fibonacci number, as the current title implies. The article mostly uses "Fibonacci numbers" or "Fibonacci sequence", not the singular "Fibonacci number". Mucube (talk • contribs) 00:41, 1 March 2023 (UTC)

• Oppose. Our articles in can be named for the individual members, in singular (Catalan number), or in some cases named for the whole sequence, in singular (Sylvester's sequence). I don't see any justification among the exceptions listed in MOS:PLURAL for doing it differently, and we certainly shouldn't do it differently for just this one article than for the rest of the category. —David Eppstein (talk) 01:38, 1 March 2023 (UTC)
• Oppose What about prime number, Catalan number, perfect number, composite number, ...? Singular seems to be the way we've done it. —Quantling (talk | contribs) 01:40, 1 March 2023 (UTC)

  Prime, composite, and perfect are extremely different from Fibonacci. While it's true that one can place the prime or composite or perfect numbers in a sequence, their defining properties do not reference the existence of the sequence or the indexing of the sequence, it is not clear a priori that they are indexed by the natural numbers (there might only be finitely many perfect numbers). "Is 233 prime?" is a meaningful question; "Is 233 Fibonacci?" is not really. --JBL (talk) 18:41, 1 March 2023 (UTC)

  Sure it is. Just check whether $5\cdot 233^{2}\pm 4$ is a square. In this case $5\cdot 233^{2}-4=521^{2}$, so yes, it is Fibonacci. Or "it is a Fibonacci number", if you prefer that grammar. In any case, it's a question you can answer for individual numbers with no reference to the sequence or its recurrence. —David Eppstein (talk) 18:49, 1 March 2023 (UTC)

  Perhaps meaningful is in the eye of the beholder. Not only is 233 a Fibonacci number, it is a Fibonacci prime and a Ramanujan prime! —Quantling (talk | contribs) 18:49, 1 March 2023 (UTC)

  The sentence "233 is a Fibonacci number" is not the same sentence as "233 is Fibonacci"; the latter is stilted and artificial. While I hold David Eppstein's opinions with respect to encyclopedia editing in high regard, I think it's safe to say that no one has ever before written the sentence "We define a positive integer f to be Fibonacci if one of $5f^{2}\pm 4$ is a perfect square". This is entirely unlike the other examples here (prime, composite, perfect), and this renders the argument by analogy unpersuasive. "A Fibonacci prime" works because of an unrelated convention about special kinds of prime numbers, not because of anything to do with Fibonacci numbers. (I happen think that moving articles from one plausible title to another is generally a bad idea and a waste of time, except when it comes to a minor personal obsession with fake royalty, so I'm not going to cast a vote in this discussion. But the analogy offered here is not at all compelling.) --JBL (talk) 19:16, 1 March 2023 (UTC)

  So numbers named after people (like Fibonacci numbers and Bell numbers) should be given a plural article title while articles whose name is an adjective (like Catalan numbers and prime numbers) should be singular, because grammar? That makes even less sense than most of the arguments here. —David Eppstein (talk) 19:32, 1 March 2023 (UTC)

  Hmm, the index of a Fibonacci number in the sequence of Fibonacci numbers is mathematically meaningful in the sense that gcd(F_i, F_j) = F_{gcd(i, j)}. I guess that that could be construed as an argument that "Fibonacci number" is too isolating. —Quantling (talk | contribs) 19:42, 1 March 2023 (UTC)

  David, I don't recognize anything to do with my point in your restatement of it. I don't know if you're intending this or if it's just how the internet makes everything worse, but your comments toward me in this sub-thread seem to fall somewhere between "snarky" and "aggressive". I have advised other editors in the past that, while I do not always agree with you, I find your perspective to be always thoughtful and worth considering; I hope it is not too much to ask a degree of consideration in return. In any case, I don't have anything further substantive to add to this discussion. --JBL (talk) 20:21, 1 March 2023 (UTC)

• Moral support. Per WP:CONSISTENT, I wouldn't want to move just this one article right now, but I agree with the logic of the nomination. In my opinion, classes of numbers which are defined by some inherent property like Prime number, Perfect number, and Composite number (where the sequence is merely a trivial ordering of the numbers which happen to satisfy that property) should be singular, but sequences (i.e. ordered lists of numbers defined by their relation to previous elements or in terms of some index variable) like Catalan numbers should be plural. I recommend proposing this at WP:NCPL and advertising it on WP:NCNUM, MOS:MATH, and WP:WPMATH. If there are any edge cases which appear to fall in both camps, we should distinguish between which property is the definition and which property is the theorem. -- King of ♥ ♦ ♣ ♠ 04:59, 1 March 2023 (UTC)

  It is easy to write property-based rather than sequence-based definitions for the Fibonacci numbers (for instance, that they are the numbers $x$ for which one of $5x^{2}\pm 4$ is square) and sequence-based rather than property-based definitions for the prime numbers (for instance, they are the sequence one gets by applying the sieve of Eratosthenes). So I don't think this distinction is particularly meaningful. And in any case, if it were the sequence rather than the numbers that were most essential here, we already have a different convention for that: name the article for the (singular) sequence. —David Eppstein (talk) 06:44, 1 March 2023 (UTC)

  I've already addressed that concern in my last point - which one is the primary definition? That might be a little subjective at times, but we deal with subjectivity at RM all the time. In this particular case, Fibonacci sequence does work and is actually more common (count me as a support for the third option), but Catalan numbers are not as commonly referred to as the Catalan sequence. For that sequence I would prefer "Catalan numbers" over "Catalan number" for the reason I described. -- King of ♥ ♦ ♣ ♠ 08:46, 1 March 2023 (UTC)

• Third option -- move to "Fibonacci Sequence" instead. It's the sequence that is the object under study, not the individual numbers. Barring that, I still support the move. David Eppstein, if you don't see applicability in WP:PLURAL, you're not looking very hard. It even calls out this very type of article: Things like Maxwell's equations, Legendre polynomials, Chebyshev polynomials, [the] Cauchy–Riemann equations, etc. The topic is naturally the set or family of equations, although in some contexts they may be referred to in the singular. (That is, such a polynomial—for example—is of interest only because it is part of the polynomial sequence called the Chebyshev polynomials, and the sequence is thought of for most purposes as a unit.) Similarly, one is much more likely to mention Arabic numerals than a particular Arabic numeral. Indeed, a Fibonacci number is only of interest because it's one of the set of Fibonacci numbers. Still though, this sequence is traditionally defined as a sequence, and not as a property that some numbers possess and others don't (like prime numbers). Quantling's analogy fails for this very reason. 35.139.154.158 (talk) 06:47, 1 March 2023 (UTC)

  Those are collections of things that don't make sense to break out of the collection and study individually. Maxwell's equations only make sense as a whole. You wouldn't generally talk about something as being "one of Maxwell's equations", by itself. It is perfectly normal to talk about an individual number like 144 as being a Fibonacci number, though, as in fact we do on 144 (number), or to talk about the properties that an individual Fibonacci number might have, not based on its position in the sequence, as we do at Fibonacci prime. That is the distinction made at MOS:PLURAL: does it make sense to talk about these individually or are they primarily a collective set? —David Eppstein (talk) 07:09, 1 March 2023 (UTC)

• Oppose per WP:COMMONNAME. The plural makes no sense, as the phrase “the Fibonacci numbers” is rarely used, since when considered as whole, the Fibonacci numbers are called the Fibonacci sequence. This is a big difference with Legendre polynomials and Chebyshev polynomials polynomials ("Legendre polynomials sequence” and "Chebyshev polynomials sequence” seems, at least, pedantic). D.Lazard (talk) 10:25, 1 March 2023 (UTC)

(edit conflict)

• Support, per WP:COMMONNAME and the literature:

Google	Scholar	Books
"Fibonacci numbers"	36.700	96.800
"Fibonacci number"	18.700	41.900
"Fibonacci sequence"	31.200	83.500

Moving to Fibonacci sequence seems ok too. - DVdm (talk) 10:33, 1 March 2023 (UTC)

• Support, as the article is about the sequence, not about individual numbers. Being a "Fibonacci number" is hardly an interesting property of an individual number. JIP | Talk 10:38, 1 March 2023 (UTC)
• Support alternate move to Fibonacci sequence so there does not need to be any more back and forth as to whether it's plural or not. ᴢxᴄᴠʙɴᴍ (ᴛ) 12:21, 1 March 2023 (UTC)
• Oppose moving to the plural, for consistency with other Wikipedia articles. I don't think raw search-engine results are a meaningful test of "commonality", and I don't think picking "the common name" is actually a meaningful standard to follow in this case. It's not like we're having to decide whether to use "Fibonacci" or "Fibonacci–Pingala", for example; the decision isn't between different terms, but different grammatical forms of the same term. XOR'easter (talk) 15:50, 1 March 2023 (UTC)
• oppose per WP:PLURAL, no problems with moving to Fibonacci sequence—blindlynx 21:02, 1 March 2023 (UTC)
• Oppose The proposed name is no better than the current one, which is already ugly, and something I haven't encountered outside Wikipedia. The primary search term is far more likely to be Fibonacci Sequence. That's what we SHOULD be moving this to. HiLo48 (talk) 23:25, 1 March 2023 (UTC)
• Oppose. The presumption that titles should be singular is not overcome. A singular Fibonacci number is sensible and not comparable to, say, a singular Maxwell's equation or a singular Cauchy–Riemann equation. Also oppose Fibonacci sequence. A phrase like Fibonacci number and its plural should be counted together for WP:COMMONNAME purposes. Adumbrativus (talk) 05:10, 7 March 2023 (UTC)

  @Adumbrativus: A paper which mentions "Fibonacci number" is also very likely to mention "Fibonacci numbers" and vice versa. So you can't just add the numbers together because the mentions are not unique. -- King of ♥ ♦ ♣ ♠ 03:44, 8 March 2023 (UTC)

• Support move to Fibonacci sequence. Favonian (talk) 18:55, 7 March 2023 (UTC)
• Support since this is a "naturally plural" concept. Also support Fibonacci sequence as second choice (I think it's a more obscure term, and arguably not the WP:COMMONNAME, though it is probably WP:RECOGNIZABLE and COMMONNAME is primarily a restatement of RECOGNIZABLE in the first place).  — SMcCandlish ☏ ¢ 😼  01:40, 9 March 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Opposition to "Fibonacci sequence" was never solicited

@ModernDayTrilobite: Of course there were very few comments posted that opposed the name "Fibonacci sequence" because that option was not on the table! If comments had been solicited on that then people would have commented and we would actually know whether it was a consensus decision. Please, if you want the title to be "Fibonacci sequence" then put it in an RfC. Yes, I know that you already renamed the page to something that wasn't under consideration, but you can reverse that. —Quantling (talk | contribs) 21:22, 10 March 2023 (UTC)

The title "Fibonacci sequence" was initially proposed on March 1, roughly nine and a half days before I closed the RM. After the initial proposition was made, virtually every participant in the RM expressed an explicit opinion on "Fibonacci sequence" as a title. In a case like this, where an alternate title is actively discussed over the course of an RM, it's very possible (and not uncommon) for that alternate title to achieve consensus. (Finally, I don't personally want any given title - it'd be a supervote if I did.) ModernDayTrilobite (talk • contribs) 14:48, 13 March 2023 (UTC)

Thank you @ModernDayTrilobite for your response. I apologize for my writing "if you want ..."; I stand corrected. Count me among those who responded before the suggestion of "Fibonacci sequence" and count me as one who would generally not get side-tracked to comment on something that wasn't solicited by the RfC. I can see value in allowing these discussions to evolve, but I'd like to see a better process for it. Perhaps one could officially amend the RfC, either at the top or in the comments at the time of its occurrence, in a highlighted, readily apparent way, to solicit comments on the new possibility. Ideally, only then should the new possibility become fair game. Thanks —Quantling (talk | contribs) 16:28, 13 March 2023 (UTC)

Cite journal

@JayBeeEll: Feel free to ignore all this below if you've not already read it; I made a mistake. – Scyrme (talk) 19:35, 9 April 2023 (UTC)

Why revert a change to "cite journal"? It's a harmless change. Setting a specific type is helpful for bots and for error checking. (For instance, if the "journal" parameter is empty it will display an error for "cite journal" but not for "citation". In this case it's not empty; it's just an example.) In all likelihood the only reason it wasn't already used was because someone was adding references of different kinds and didn't want to figure out which references needed which template.

• Douady, S; Couder, Y (1996), "Phyllotaxis as a Dynamical Self Organizing Process" (PDF), Journal of Theoretical Biology, 178 (3): 255–74, doi:10.1006/jtbi.1996.0026, archived from the original (PDF) on 2006-05-26

• Douady, S; Couder, Y (1996), "Phyllotaxis as a Dynamical Self Organizing Process" (PDF), Journal of Theoretical Biology, 178 (3): 255–74, doi:10.1006/jtbi.1996.0026, archived from the original (PDF) on 2006-05-26

The first uses "cite journal", the second uses "citation". All the information is in the same order. – Scyrme (talk) 18:40, 9 April 2023 (UTC)

{{cite journal}} is Citation Style 1. {{citation}} is Citation Style 2. They differ. Most obviously in the. Way that. Citation Style 1. Breaks things up. With lots. Of. Periods. WP:CITEVAR is very clear that you should not be changing citation styles in this way without consensus. For those of us who use User:BrandonXLF/CitationStyleMarker.js to find inconsistent citation styles, your change is very annoying because it causes all of the citations to be flagged as inconsistent. Also your claim that this is helpful for bots and error checking seems dubious to me. The bots all know how to deal with both styles. —David Eppstein (talk) 18:45, 9 April 2023 (UTC)

The documentation indicates that this can be toggled with a parameter. Setting |mode=cs2 should make them identical. – Scyrme (talk) 18:50, 9 April 2023 (UTC)

@David Eppstein: Regarding bots, I've seen some change the type (from "web" to "journal" or whatever, not CS1 to CS2), so I assumed that it would reduce the task queue if nothing else. I wasn't aware that {{citation}} used a different style by default to the other citation templates. Surprises me that it does, to be honest. That seems needlessly confusing.

Does setting the "mode" parameter still result in that annoyance? – Scyrme (talk) 18:57, 9 April 2023 (UTC)

Not that specific annoyance. But if you keep going around making cosmetic changes to the source code of articles without doing anything actually constructive, it will still continue to annoy me by cluttering my watchlist and wasting my time checking the edits. There's also an argument that it's still in violation of CITEVAR, because CITEVAR controls the appearance of citation sourcing (for instance, whether the references are list-defined or inline), not just the readers' view of citations. The article currently uses one template consistently for its citations, and you would be going against that consistency. Changing cite web to cite journal is different: that's staying within the same family of templates and merely (one hopes) making the type more accurate. —David Eppstein (talk) 19:03, 9 April 2023 (UTC)

Like I said, I wasn't aware that {{citation}} wasn't in the same family of templates. I thought I was just making the template more accurate. Now that I know, I won't mix them up. – Scyrme (talk) 19:24, 9 April 2023 (UTC)

Every k-th Fibonacci number

Does anyone have a reference to (or knowledge of) linear recurrence formulas for every-k-th term of the Fibonacci sequence? For example with k = 2, we can compute every other value recursively with F_n+4 = 3F_n+2 − F_n, for any integer n ≥ 0. Likewise with k = 3, we can compute every third value F_n+6 = 4F_n+3 + F_n. With k = 4, we can compute every fourth value with F_n+8 = 7F_n+4 − F_n. —Quantling (talk | contribs) 15:01, 10 May 2023 (UTC)

Apply the roots of unity filter to the GF to get the GF for the multisection. The denominator will be $\prod _{i=0}^{k-1}(1-\omega ^{k}x-\omega ^{2k}x^{2})$ where $\omega$ is a primitive complex k-th root of unity, and its coefficients are the coefficients of the recurrence (by general theory of when ordinary GFs are rational functions). --JBL (talk) 17:55, 10 May 2023 (UTC)

It seems to be the case that

$${\begin{aligned}F_{n+2k}&=\alpha F_{n+k}+\beta F_{n}\mathrm {,with} \\\alpha &={\frac {F_{2k}}{F_{k}}}\\\beta &=F_{2k+1}-{\frac {F_{2k}}{F_{k}}}F_{k+1}\,.\end{aligned}}$$

But, even if this qualifies as simple calculation, we are going to need proof of notability. —Quantling (talk | contribs) 17:55, 10 May 2023 (UTC)

Those coefficients are a Lucas number and $\pm 1$. Personally I would not be inclined to include this for reasons of due weight. --JBL (talk) 18:00, 10 May 2023 (UTC)

Thank you. So that makes it

One can compute every k-th Fibonacci number with the recurrence F_n+2k = L_k F_n+k − (−1)^k F_n, where n is any non-negative integer and L_k = F_k+1 + F_k−1 is a Lucas number.

That simplifies the presentation. (Still need indication that it has due weight.) —Quantling (talk | contribs) 18:29, 10 May 2023 (UTC)

I am going ahead with a WP:BRD edit, with the hope that it draws additional editors to this discussion. —Quantling (talk | contribs) 18:44, 11 May 2023 (UTC)

You added it to the article without any source at all. Please do not do that. It needs a published source. —David Eppstein (talk) 20:18, 11 May 2023 (UTC)

If someone has a way to search back issues of Fibonacci Quarterly, there's absolutely no way this hasn't been published before. I'm not at all worried about this being unverifiable. But even if sources are hunted down, I still wonder whether/why this identity is noteworthy among the dozens of similar identities that could be written down. –jacobolus (t) 05:14, 12 May 2023 (UTC)

For example, it's exercise 52–53 in https://doi.org/10.1002/9781118033067.ch5 and theorems 5, 7 in https://doi.org/10.2307/3619903 –jacobolus (t) 09:51, 12 May 2023 (UTC)

If we can find discussion in Fibonacci Quarterly, perhaps that will signal noteworthiness as well. Unfortunately, I do not have access. —Quantling (talk | contribs) 13:28, 12 May 2023 (UTC)

You see the Math. Gazette article right?

Koshy, Thomas (1998). "82.55 New Fibonacci and Lucas Identities". Mathematical Gazette. 82 (495): 481–484. doi:10.2307/3619903. JSTOR 3619903.

It has Theorems 4–7:

${\begin{aligned}L_{2m+n}-(-1)^{m}L_{n}&=5F_{m}F_{m+n},\\[3mu]F_{2m+n}-(-1)^{m}F_{n}&=F_{m}L_{m+n},\\[3mu]L_{2m+n}+(-1)^{m}L_{n}&=L_{m}L_{m+n},\\[3mu]F_{2m+n}+(-1)^{m}F_{n}&=L_{m}F_{m+n}.\end{aligned}}$

But the main question is: why do you think this identity is important out of the dozens of similar identities one could write down? –jacobolus (t) 15:23, 12 May 2023 (UTC)

Why does it appeal to me?: because I find it interesting that a sequence that is every k-th term of the Fibonacci sequence obeys a linear recurrence (and that the linear recurrence has easily computed coefficients). Of the above, only Theorem 7 is about such a sequence. While I find pretty much all math to be at least somewhat interesting, I don't (yet) see as much beauty in the other three theorems. FWIW, there are formulas that are already in the article that I also find less interesting than this subsequence formula.

But obviously this isn't about me, and unfortunately I don't have access to pretty much anything proprietary online. If there is an established author who shows interest similar to mine then I am hopeful that we could build consensus around including this formula. —Quantling (talk | contribs) 15:49, 12 May 2023 (UTC)

Do you know about the "Wikipedia Library"? –jacobolus (t) 18:50, 12 May 2023 (UTC)

No. Nice. —Quantling (talk | contribs) 19:49, 12 May 2023 (UTC)

It looks both @JBL and @jacobolus are against including this text for reasons of due weight. Absent new evidence, it looks like the proposed edit is not going happen. FWIW, maybe @David Eppstein's concern for a reference has been satisfied. Thank you for the discussion —Quantling (talk | contribs) 13:56, 16 May 2023 (UTC)

I am ambivalent about it. I haven't thought much at all about this particular article. But in general, it's hard to figure out which identities are interesting enough to tell readers about. –jacobolus (t) 14:52, 16 May 2023 (UTC)

One other highly cited paper you may want to take a look at is Rabinowitz (1996) doi:10.1007/978-94-009-0223-7_33 http://stanleyrabinowitz.com/download/algorithmicfib.pdf –jacobolus (t) 19:35, 16 May 2023 (UTC)

Fibonacci Approximation Function

A function that approximates the Fibonacci sequence is the following: y = (0.003)x^6 - (0.0064)x^5 + (0.0701)x^4 - (0.3435)x^3 + (1.0052)x^2 -(0.6578)x + 0.9895

Try and plot it on Octave or in any other software.

~_Roberto::Barone_~ 95.234.175.68 (talk) 15:14, 18 September 2023 (UTC)

It is possible to come up with many polynomials that accurately approximate a few of the Fibonacci numbers. (See polynomial fitting, generally.) However, regardless of the polynomial p(x), it is the case that

$$\lim _{x\rightarrow +\infty }{\frac {p(x)}{\phi ^{x}}}=0\,,$$

and thus the polynomial will fall woefully short of the Fibonacci sequence for large enough values of x. —Quantling (talk | contribs) 15:38, 18 September 2023 (UTC)

Thanks for your reply.

I know... Mine was just an approximate attempt for the sequence.

~_Roberto::Barone_~ 95.234.175.68 (talk) 16:01, 18 September 2023 (UTC)

Pictures in article

The first two pictures in the article illustrate some features of the squares of Fibonacci numbers. Don't we have any pictures to illustrate the actual Fibonacci numbers instead of their squares? Eddi (Talk) 16:32, 2 October 2023 (UTC)