Octagonal number

Last updated January 07, 2025

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

Applications in combinatorics

The $n$ th octagonal number is the number of partitions of $6n-5$ into 1, 2, or 3s.^[2] 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^[3] $\sum _{n=1}^{\infty }{\frac {1}{n(3n-2)}}={\frac {9\ln(3)+{\sqrt {3}}\pi }{12}}.$

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.

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<span class="mw-page-title-main">Tetrahedral number</span> Polyhedral number representing a tetrahedron

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References

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

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[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.

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