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{{refimprove|date=October 2013}}
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An '''octagonal number''' is a [[figurate number]] that represents an [[octagon]]. 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, [[280 (number)|280]], 341, 408, 481, 560, 645, 736, 833, 936 {{OEIS|id=A000567}}
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Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the ''n''th octagonal number is
 
<math>n^2 + 4\sum_{k = 1}^{n - 1} k = 3n^2-2n</math>
 
which simplifies to the formulas given above.
 
The octagonal number for ''n'' can also be calculated by adding the square of ''n'' to twice the (''n - 1'')th [[pronic number]], or, to put it algebraically, <math>O_n = 3n^2-2n</math>.
 
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|first2=Michel|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=57|url=http://books.google.com/books?id=cDxYdstLPz4C&pg=PA57}}.</ref>
 
==See also==
* [[Centered octagonal number]]
 
==References==
{{reflist}}
 
{{Classes of natural numbers}}
[[Category:Figurate numbers]]
 
 
{{num-stub}}

Latest revision as of 20:49, 12 January 2015

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