Decagonal number

From Wikipedia, the free encyclopedia

A decagonal number is a figurate number that represents a decagon. The decagonal number for n is given by the formula

4n² − 3n

with n > 0. The first few decagonal numbers are

1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326, 8695, 9072, 9457, 9850 (sequence A001107 in OEIS)

The decagonal number for n can also be calculated by adding the square of n to thrice the (n - 1)th pronic number, or to put it algebraically, D_n = n² + 3(n² − n).

Decagonal numbers consistently alternate parity.