View source for Octagonal number

{{Short description|Number of points in an octagonal arrangement}}
{{more citations needed|date=October 2013}}
[[File:OctagonalNumbers.svg|thumb|The first five octagonal numbers illustrated.]]
In [[mathematics]], an '''octagonal number''' is a [[figurate number]]. The ''n''th octagonal number ''o''<sub>''n''</sub> 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 (geometry)|vertex]]. The octagonal number for ''n'' is given by the formula 3''n''<sup>2</sup> − 2''n'', with ''n'' > 0. The first few octagonal numbers are

: [[1 (number)|1]], [[8 (number)|8]], [[21 (number)|21]], [[40 (number)|40]], [[65 (number)|65]], [[96 (number)|96]], [[133 (number)|133]], [[176 (number)|176]], [[225 (number)|225]], [[280 (number)|280]], 341, 408, 481, 560, 645, 736, 833, 936 {{OEIS|id=A000567}}

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 (mathematics)|parity]].

Octagonal numbers are occasionally referred to as "[[star number]]s", though that term is more commonly used to refer to centered dodecagonal numbers.<ref>{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|author1-link=Elena Deza|first2=Michel|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=57|url=https://books.google.com/books?id=cDxYdstLPz4C&pg=PA57}}.</ref>

==Applications in combinatorics==
The <math>n</math>th octagonal number is the number of [[integer partition|partitions]] of <math>6n-5</math> into 1, 2, or 3s.<ref>{{OEIS|id=A000567}}</ref> For example, there are <math>x_2=8</math> such partitions for <math>2\cdot 6-5=7</math>, 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 [[sums of reciprocals|sum of the reciprocals]] of the octagonal numbers is given by<ref>{{Cite web |url=http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |title=Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers |access-date=2020-04-12 |archive-date=2013-05-29 |archive-url=https://web.archive.org/web/20130529032918/http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |url-status=dead }}</ref>
<math display=block>
\sum_{n=1}^\infty \frac{1}{n(3n-2)} = \frac{9\ln(3)+\sqrt3\pi}{12}.
</math>

==Test for octagonal numbers==

Solving the formula for the ''n''-th octagonal number, <math>x_n,</math> for ''n'' gives
<math display=block>n= \frac{\sqrt{3x_n+1}+1}{3}.</math>
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==
{{reflist}}

{{Figurate numbers}}
{{Classes of natural numbers}}

[[Category:Figurate numbers]]


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