An octagonal number is a figurate number that represents an octagon. The octagonal number for n is given by the formula 3n2 - 2n, with n > 0. The first few octagonal numbers are:
1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936 (sequence A000567 in OEIS)
Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the nth octagonal number is
which simplifies to the formula given above.
The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number, or, to put it algebraically, On = n2 + 2(n2 − n).
Octagonal numbers consistently alternate parity.
Confusingly, octagonal numbers are occasionally referred to as "star numbers," though that term is more commonly used to refer to centered dodecagonal numbers.