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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.

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 $\frac{n^2[s-2]-n[s-4]}{2}$.

 Name Formula n=1 2 3 4 5 6 7 8 9 10 11 12 13 Triangular ½(n² + n) 1 3 6 10 15 21 28 36 45 55 66 78 91 Square n² 1 4 9 16 25 36 49 64 81 100 121 144 169 Pentagonal ½(3n² - n) 1 5 12 22 35 51 70 92 117 145 176 210 247 Hexagonal 2n² - n 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 3n² - 2n 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 (2n)² - 3n 1 10 27 52 85 126 175 232 297 370 451 540 637 Hendecagonal ½((3n)² - 7n) 1 11 30 58 95 141 196 260 333 415 506 606 715 Dodecagonal 5n² - 4n 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 6n² - 5n 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 7n² - 6n 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 8n² - 7n 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 (3n)² - 8n 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 10n² - 9n 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 11n² - 10n 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 12n² - 11n 1 26 75 148 245 366 511 680 873 1090 1331 1596 1885 Icosiheptagonal ½((5n)² - 23n) 1 27 78 154 255 381 532 708 909 1135 1386 1662 1963 Icosioctagonal 13n² - 12n 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 14n² - 13n 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

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