Octagonal number

The first five octagonal numbers illustrated.

In mathematics, an octagonal number is a figurate number. The nth octagonal number o_n is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to n dots, when the octagons are overlaid so that they share one vertex. 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. ^[1]

Applications in combinatorics

The ${\displaystyle n}$th octagonal number is the number of partitions of ${\displaystyle 6n-5}$ into 1, 2, or 3s. ^[2] For example, there are ${\displaystyle x_{2}=8}$ such partitions for ${\displaystyle 2\cdot 6-5=7}$, 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 ^[3] $${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(3n-2)}}={\frac {9\ln(3)+{\sqrt {3}}\pi }{12}}.}$$

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.

See also

• Centered octagonal number

References

1. Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, p. 57, ISBN   9789814355483 .
2. (sequence A000567 in the OEIS )
3. "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers" (PDF). Archived from the original (PDF) on 2013-05-29. Retrieved 2020-04-12.