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 n-th octagonal number is

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

The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number.

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.[1]

## Sum of reciprocals

A formula for the sum of the reciprocals of the octagonal numbers is given by:[2]

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(3n-2)))={\frac {9\ln(3)+{\sqrt {3))\pi }{12))}$

## Test for octagonal numbers

Solving the formula for the n-th octagonal number, ${\displaystyle x_{n},}$ for n gives

${\displaystyle n={\frac ((\sqrt {3x_{n}+1))+1}{3)).}$

An arbitrary number x can be checked for octagonality by putting it in this equation. If n is an integer, then x is the n-th octagonal number. If n is not an integer, then x is not octagonal.