Polygonal number

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In mathematics, a polygonal number is a number represented as dots or pebbles arrayed in the shape of a polygon. The dots were thought of as alphas (units). These are one type of figurate numbers.

Contents

[edit] Definition and examples

The number 10, for example, can be arranged as a triangle (see triangular number):

*
**
***
****

But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):

***
***
***

Some numbers, like 36, can be arranged both as a square and as a triangle (see triangular square number):

******
******
******
******
******
******		 *
**
***
****
*****
******
*******
********

By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red.

Triangular numbers
1 3 6 10
* *
**		 *
**
***		 *
**
***
****
Square numbers
1 4 9 16
* **
**		 ***
***
***		 ****
****
****
****

Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a regular lattice like above. For example, the first few hexagonal numbers are:

1 6 15 28
* **
* *
**		 ***
** *
* * *
** *
***		 ****
*** *
** * *
* * * *
** * *
*** *
****

If s is the number of sides in a polygon, the formula for the nth s-gonal number is {(\frac{s}{2}-1)n^2-(\frac{s}{2}-2)n}.

Name Formula n=1 2 3 4 5 6 7 8 9 10 11 12 13 Sum of Reciprocals[1] OEIS number
Triangular ½(1n² + 1n) 1 3 6 10 15 21 28 36 45 55 66 78 91 2 A000217
Square ½(2n² - 0n) 1 4 9 16 25 36 49 64 81 100 121 144 169 {\pi^2 \over 6} A000290
Pentagonal ½(3n² - 1n) 1 5 12 22 35 51 70 92 117 145 176 210 247 3\ln\left(3\right)-{\pi\sqrt{3}\over3} A000326
Hexagonal ½(4n² - 2n) 1 6 15 28 45 66 91 120 153 190 231 276 325 2\ln\left(2\right) A000384
Heptagonal ½(5n² - 3n) 1 7 18 34 55 81 112 148 189 235 286 342 403 A000566
Octagonal ½(6n² - 4n) 1 8 21 40 65 96 133 176 225 280 341 408 481 {3\ln\left(3\right)\over 4}+{\sqrt{3}\pi\over 12} A000567
Nonagonal ½(7n² - 5n) 1 9 24 46 75 111 154 204 261 325 396 474 559 A001106
Decagonal ½(8n² - 6n) 1 10 27 52 85 126 175 232 297 370 451 540 637 2\ln\left(2\right)+{\pi \over 6} A001107
Hendecagonal ½(9n² - 7n) 1 11 30 58 95 141 196 260 333 415 506 606 715 A051682
Dodecagonal ½(10n² - 8n) 1 12 33 64 105 156 217 288 369 460 561 672 793 A051624
Tridecagonal ½(11n² - 9n) 1 13 36 70 115 171 238 316 405 505 616 738 871 A051865
Tetradecagonal ½(12n² - 10n) 1 14 39 76 125 186 259 344 441 550 671 804 949 {2\ln\left(2\right)\over 5}+{3\ln\left(3\right)\over 10}+{\sqrt{3}\pi\over 10} A051866
Pentadecagonal ½(13n² - 11n) 1 15 42 82 135 201 280 372 477 595 726 870 1027 A051867
Hexadecagonal ½(14n² - 12n) 1 16 45 88 145 216 301 400 513 640 781 936 1105 A051868
Heptadecagonal ½(15n² - 13n) 1 17 48 94 155 231 322 428 549 685 836 1002 1183 A051869
Octadecagonal ½(16n² - 14n) 1 18 51 100 165 246 343 456 585 730 891 1068 1261 A051870
Nonadecagonal ½(17n² - 15n) 1 19 54 106 175 261 364 484 621 775 946 1134 1339 A051871
Icosagonal ½(18n² - 16n) 1 20 57 112 185 276 385 512 657 820 1001 1200 1417 A051872
Icosihenagonal ½(19n² - 17n) 1 21 60 118 195 291 406 540 693 865 1056 1266 1495 A051873
Icosidigonal ½(20n² - 18n) 1 22 63 124 205 306 427 568 729 910 1111 1332 1573 A051874
Icositrigonal ½(21n² - 19n) 1 23 66 130 215 321 448 596 765 955 1166 1398 1651 A051875
Icositetragonal ½(22n² - 20n) 1 24 69 136 225 336 469 624 801 1000 1221 1464 1729 A051876
Icosipentagonal ½(23n² - 21n) 1 25 72 142 235 351 490 652 837 1045 1276 1530 1807
Icosihexagonal ½(24n² - 22n) 1 26 75 148 245 366 511 680 873 1090 1331 1596 1885
Icosiheptagonal ½(25n² - 23n) 1 27 78 154 255 381 532 708 909 1135 1386 1662 1963
Icosioctagonal ½(26n² - 24n) 1 28 81 160 265 396 553 736 945 1180 1441 1728 2041
Icosinonagonal ½(27n² - 25n) 1 29 84 166 275 411 574 764 981 1225 1496 1794 2119
Triacontagonal ½(28n² - 26n) 1 30 87 172 285 426 595 792 1017 1270 1551 1860 2197

The On-Line Encyclopedia of Integer Sequences eschews terms using Greek prefixes (e.g., "octagonal") in favor of terms using numerals (i.e., "8-gonal").

For a given s-gonal number x, one can find n by

n = \frac{\sqrt{(8s-16)x+(s-4)^2}+s-4}{2s-4}.

[edit] Combinations

Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets. The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to Pell's equation. The simplest example of this is the sequence of square triangular numbers.

The following table summarizes the set of s-gonal t-gonal numbers for small values of s and t.

s t Sequence OEIS number
4 3 1, 36, 1225, 41616, … A001110
5 3 1, 210, 40755, 7906276, … A014979
5 4 1, 9801, 94109401, … A036353
6 3 All hexagonal numbers are also triangular. A000384
6 4 1, 1225, 1413721, 1631432881, … A046177
6 5 1, 40755, 1533776805, … A046180
7 3 1, 55, 121771, 5720653, … A046194
7 4 1, 81, 5929, 2307361, … A036354
7 5 1, 4347, 16701685, 64167869935, … A048900
7 6 1, 121771, 12625478965, … A048903
8 3 1, 21, 11781, 203841, … A046183
8 4 1, 225, 43681, 8473921, … A036428
8 5 1, 176, 1575425, 234631320, … A046189
8 6 1, 11781, 113123361, … A046192
8 7 1, 297045, 69010153345, … A048906
9 3 1, 325, 82621, 20985481, … A048909
9 4 1, 9, 1089, 8281, 978121, … A036411
9 5 1, 651, 180868051, … A048915
9 6 1, 325, 5330229625, … A048918
9 7 1, 26884, 542041975, … A048921
9 8 1, 631125, 286703855361, … A048924

In some cases, such as s=10 and t=4, there are no numbers in both sets other than 1.

The problem of finding numbers that belong to three polygonal sets is more difficult. A computer search for pentagonal square triangular numbers has yielded only the trivial value of 1, though a proof that there are no such number has yet to appear in print.[2] All hexagonal square numbers are also hexagonal square triangular numbers, and 1225 is actually a hecticositetragonal, hexacontagonal, icosinonagonal, hexagonal, square, triangular number.

[edit] Notes

  1. ^ Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers
  2. ^ Weisstein, Eric W., "Pentagonal Square Triangular Number" from MathWorld.

[edit] References

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Polygonal_number"