Pentagonal number

Encyclopedia : P : PE : PEN : Pentagonal number

A pentagonal number is a figurate number that represents a pentagon. The nth pentagonal number p_n is given by the formula:

[p_n = \frac2]

for n ≥ 1. 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 (sequence in OEIS)

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

The nth pentagonal number is one third of the 3n-1th triangular number.

Pentagonal numbers should not be confused with centered pentagonal numbers.

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