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 n-th octagonal number is

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

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.

Sum of reciprocals

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

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

Test for octagonal numbers

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

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

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.

See also

• Centered octagonal number