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

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

$n^2\; +\; 4\backslash sum\_\{k\; =\; 1\}^\{n\; -\; 1\}\; k$

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, $O\_n\; =\; n^2\; +\; 2(n^2\; -\; n)$.

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.

Figurate numbers

Nombre octogonal | Numero ottagonale | Kahdeksankulmioluku