| [Edit]
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; such numbers, which can be made from figures, are generally called figurate numbers.
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
Square numbers
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:
If s is the number of sides in a polygon, the formula for the nth s-gonal number is .
Name | Formula | n=1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Triangular | ½(1n² + 1n) |
1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 |
| Square | ½(2n² - 0n) |
1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 |
| Pentagonal | ½(3n² - 1n) |
1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | 176 | 210 | 247 |
| Hexagonal | ½(4n² - 2n) |
1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | 231 | 276 | 325 |
| Heptagonal | ½(5n² - 3n) |
1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | 286 | 342 | 403 |
| Octagonal | ½(6n² - 4n) |
1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | 341 | 408 | 481 |
| Nonagonal | ½(7n² - 5n) |
1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | 396 | 474 | 559 |
| Decagonal | ½(8n² - 6n) |
1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | 451 | 540 | 637 |
| Hendecagonal | ½(9n² - 7n) |
1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | 506 | 606 | 715 |
| Dodecagonal | ½(10n² - 8n) |
1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | 561 | 672 | 793 |
| Tridecagonal | ½(11n² - 9n) |
1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | 616 | 738 | 871 |
| Tetradecagonal | ½(12n² - 10n) |
1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | 671 | 804 | 949 |
| Pentadecagonal | ½(13n² - 11n) |
1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | 726 | 870 | 1027 |
| Hexadecagonal | ½(14n² - 12n) |
1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | 781 | 936 | 1105 |
| Heptadecagonal | ½(15n² - 13n) |
1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | 836 | 1002 | 1183 |
| Octadecagonal | ½(16n² - 14n) |
1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | 891 | 1068 | 1261 |
| Nonadecagonal | ½(17n² - 15n) |
1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | 946 | 1134 | 1339 |
| Icosagonal | ½(18n² - 16n) |
1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | 1001 | 1200 | 1417 |
| Icosihenagonal | ½(19n² - 17n) |
1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | 1056 | 1266 | 1495 |
| Icosidigonal | ½(20n² - 18n) |
1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | 1111 | 1332 | 1573 |
| Icositrigonal | ½(21n² - 19n) |
1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | 1166 | 1398 | 1651 |
| Icositetragonal | ½(22n² - 20n) |
1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | 1221 | 1464 | 1729 |
| 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").
|
|
