The Pentagonal number reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Pentagonal number

A pentagonal number is a figurate number that represents a pentagon. The pentagonal number for n is given by the formula n(3n - 1)/2, with n > 0. The first few pentagonal numbers are

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001

Every pentagonal number for n is one third of the triangular number for n.

Pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem.

"Generalized" pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, -1, 2, -2, 3, -3, 4..., producing the sequence

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027

Pentagonal numbers should not be confused with centered pentagonal numbers.

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