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:p_n = rac{n(3n-1)}2

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

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 ''n''th pentagonal number is one third of the 3''n''-1th Triangular Number .

Pentagonal numbers should not be confused with Centered Pentagonal Number s.