Pentagonal number
From Free net encyclopedia
A pentagonal number is a figurate number that represents a pentagon. The nth pentagonal number pn is given by the formula:
- <math>p_n = \frac{n(3n-1)}2</math>
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 Template: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.
de:Fünfeckszahl
fr:Nombre pentagonal it:Numero pentagonale sl:Peterokotniško število zh:五角数