Decagonal number

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

$D_{n}=4n^{2}-3n.$

The first few decagonal numbers are:

0, 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 (sequence A001107 in the OEIS)

The n-th decagonal number can also be calculated by adding the square of n to thrice the (n—1)-th pronic number or, to put it algebraically, as

$D_{n}=n^{2}+3(n^{2}-n).$

Properties

• Decagonal numbers consistently alternate parity.