Octagonal number

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 the 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

$n^{2}+4\sum _{k=1}^{n-1}k=3n^{2}-2n$

which simplifies to the formulas 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, $O_{n}=3n^{2}-2n$.

Octagonal numbers consistently alternate parity.

Octagonal numbers are occasionally referred to as "star numbers," though that term is more commonly used to refer to centered dodecagonal numbers.

See also

• Centered octagonal number