Multi-Polygonal numbers

Polygonal number families, formulas, and first values

General formula

Polygonal numbers count dots that can be arranged into regular polygons. For an s-gonal family, the general formula is:

Ps(n) = ((s-2)n2 - (s-4)n)/2

where s ≥ 3 (sides), n ∈ ℕ, n ≥ 1.

Source: Polygonal number (Wikipedia)

Triangular

s = 3 P3(n) = n(n + 1)/2

P3(4) = 10

First values
1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210

Square

s = 4 P4(n) = n2

P4(4) = 16

First values
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400

Pentagonal

s = 5 P5(n) = n(3n − 1)/2

P5(3) = 12

First values
1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 376 425 477 532 590

Hexagonal

s = 6 P6(n) = n(2n − 1)

P6(3) = 15

First values
1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 496 561 630 703 780

Heptagonal

s = 7 P7(n) = n(5n − 3)/2

P7(3) = 18

First values
1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 616 697 783 874 970

Octogonal

s = 8 P8(n) = n(3n − 2)

P8(3) = 21

First values
1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 736 833 936 1045 1160

Nonagonal

s = 9 P9(n) = n(7n − 5)/2

P9(3) = 24

First values
1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 856 969 1089 1216 1350

Decagonal

s = 10 P10(n) = n(4n − 3)

P10(3) = 27

First values
1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 976 1105 1242 1387 1540