Octagonal number explained

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

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]

Applications in combinatorics

The

n th octagonal number is the number of partitions of 6 n -5 into 1, 2, or 3s. For example, there are x _{2=8} such partitions for 2 ⋅ 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^[2] $$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac.$$

Test for octagonal numbers

Solving the formula for the n-th octagonal number,

x _{n,} for n gives$$n=\; \backslash frac.$$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

Notes and References

1. .
2. Web site: Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers . 2020-04-12 . 2013-05-29 . https://web.archive.org/web/20130529032918/http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf . dead .