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In mathematics, a polygonal number is a number that can be arranged as a regular polygon. Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds. The number 10, for example, can be arranged as a triangle (see triangular number):
x
x x
x x x
x x x x
But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):
x x x
x x x
x x x
Some numbers, like 36, can be arranged both as a square and as a triangle (see triangular square number):
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x
x x
x x x
x x x x
x x x x x
x x x x x x
x x x x x x x
x x x x x x x x
The method 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 +.
Triangular numbers
1:
+ x
3:
x x
+ + x x
6:
x x
x x x x
+ + + x x x
10:
x x
x x x x
x x x x x x
+ + + + x x x x
Square numbers
1:
+ x
4:
x + x x
+ + x x
9:
x x + x x x
x x + x x x
+ + + x x x
16:
x x x + x x x x
x x x + x x x x
x x x + x x x x
+ + + + x x x x
Polygons with higher numbers of sides, such as pentagons and hexagons, can also be represented as arrangements of dots (by convention 1 is the first polygonal number for any number of sides).
Pentagonal numbers:
1:
+ x
5:
x x
+ + x x
+ + x x
12:
x x
x x x x
+ x x + x x x x
+ + x x
+ + + x x x
22:
x x
x x x x
x x x x x x x x
+ x x + x x x x
+ x x x + x x x x x
+ + x x
+ + + + x x x x
35:
x x
x x x x
x x x x x x x x
x x x x x x x x
+ x x x x x + x x x x x x x
+ x x + x x x x
+ x x x x + x x x x x x
+ + x x
+ + + + + x x x x x
Hexagonal numbers
1:
x
6:
x x
+ + x x
+ + x x
+ x
15:
x x
x x x x
+ x x + x x x x
+ x + x x x
+ + x x
+ + x x
+ x
28:
x x
x x x x
x x x x x x x x
+ x x x + x x x x x
+ x x + x x x x
+ x x + x x x x
+ x + x x x
+ + x x
+ + x x
+ x
45:
x x
x x x x
x x x x x x x x
x x x x x x x x x x
+ x x x x + x x x x x x
+ x x x x + x x x x x x
+ x x x + x x x x x
+ x x + x x x x
+ x x + x x x x
+ x + x x x
+ + x x
+ + x x
+ x
66: (which is also a triangular number and a sphenic number)
x x
x x x x
x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x x x
+ x x x x x x + x x x x x x x x
+ x x x x x + x x x x x x x
+ x x x x + x x x x x x
+ x x x x + x x x x x x
+ x x x + x x x x x
+ x x + x x x x
+ x x + x x x x
+ x + x x x
+ + x x
+ + x x
+ x
91:
x x
x x x x
x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x x x x x
+ x x x x x x x + x x x x x x x x x
+ x x x x x x + x x x x x x x x
+ x x x x x x + x x x x x x x x
+ x x x x x + x x x x x x x
+ x x x x + x x x x x x
+ x x x x + x x x x x x
+ x x x + x x x x x
+ x x + x x x x
+ x x + x x x x
+ x + x x x
+ + x x
+ + x x
+ x
If s is the number of sides in a polygon, the formula for the nth s-polygonal number is ½n((s-2)n - (4-s)).
| Name | Formula | n=1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Triangular | ½n(1n + 1) |
1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 |
| Square | ½n(2n - 0) |
1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 |
| Pentagonal | ½n(3n - 1) |
1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | 176 | 210 | 247 |
| Hexagonal | ½n(4n - 2) |
1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | 231 | 276 | 325 |
| Heptagonal | ½n(5n - 3) |
1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | 286 | 342 | 403 |
| Octagonal | ½n(6n - 4) |
1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | 341 | 408 | 481 |
| Nonagonal | ½n(7n - 5) |
1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | 396 | 474 | 559 |
| Decagonal | ½n(8n - 6) |
1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | 451 | 540 | 637 |
| 11-gonal | ½n(9n - 7) |
1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | 506 | 606 | 715 |
| 12-gonal | ½n(10n - 8) |
1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | 561 | 672 | 793 |
| 13-gonal | ½n(11n - 9) |
1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | 616 | 738 | 871 |
| 14-gonal | ½n(12n - 10) |
1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | 671 | 804 | 949 |
| 15-gonal | ½n(13n - 11) |
1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | 726 | 870 | 1027 |
| 16-gonal | ½n(14n - 12) |
1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | 781 | 936 | 1105 |
| 17-gonal | ½n(15n - 13) |
1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | 836 | 1002 | 1183 |
| 18-gonal | ½n(16n - 14) |
1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | 891 | 1068 | 1261 |
| 19-gonal | ½n(17n - 15) |
1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | 946 | 1134 | 1339 |
| 20-gonal | ½n(18n - 16) |
1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | 1001 | 1200 | 1417 |
| 21-gonal | ½n(19n - 17) |
1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | 1056 | 1266 | 1495 |
| 22-gonal | ½n(20n - 18) |
1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | 1111 | 1332 | 1573 |
| 23-gonal | ½n(21n - 19) |
1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | 1166 | 1398 | 1651 |
| 24-gonal | ½n(22n - 20) |
1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | 1221 | 1464 | 1729 |
| 25-gonal | ½n(23n - 21) |
1 | 25 | 72 | 142 | 235 | 351 | 490 | 652 | 837 | 1045 | 1276 | 1530 | 1807 |
| 26-gonal | ½n(24n - 22) |
1 | 26 | 75 | 148 | 245 | 366 | 511 | 680 | 873 | 1090 | 1331 | 1596 | 1885 |
| 27-gonal | ½n(25n - 23) |
1 | 27 | 78 | 154 | 255 | 381 | 532 | 708 | 909 | 1135 | 1386 | 1662 | 1963 |
| 28-gonal | ½n(26n - 24) |
1 | 28 | 81 | 160 | 265 | 396 | 553 | 736 | 945 | 1180 | 1441 | 1728 | 2041 |
| 29-gonal | ½n(27n - 25) |
1 | 29 | 84 | 166 | 275 | 411 | 574 | 764 | 981 | 1225 | 1496 | 1794 | 2119 |
| 30-gonal | ½n(28n - 26) |
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 numberals (i.e., "8-gonal").
References
sl:mnogokotniško število
he:מספר פוליגונלי
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