Octagonal number

An octagonal number is a figurate number that gives the number of points in a certain octagonal arrangement. 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)

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.

Applications in combinatorics

The $n$th octagonal number is the number of partitions of $6n-5$ into 1, 2, or 3s. For example, there are $x_{2}=8$ such partitions for $2\cdot 6-5=7$, namely

[1,1,1,1,1,1,1], [1,1,1,1,1,2], [1,1,1,1,3], [1,1,1,2,2], [1,1,2,3], [1,2,2,2], [1,3,3] and [2,2,3].

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