Editing Centered octagonal number

{{Short description|Centered figurate number that represents an octagon with a dot in the center}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
[[Image:Centered octagonal number.svg|200px|right]]

A '''centered octagonal number''' is a [[centered number|centered]] [[figurate number]] that represents an [[octagon]] with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.<ref>{{citation
 | last1 = Teo | first1 = Boon K.
 | last2 = Sloane | first2 = N. J. A. | author2-link = Neil Sloane
 | journal = Inorganic Chemistry
 | pages = 4545–4558
 | title = Magic numbers in polygonal and polyhedral clusters
 | url = http://neilsloane.com/doc/magic1/magic1.pdf
 | volume = 24
 | issue = 26
 | year = 1985 | doi=10.1021/ic00220a025}}.</ref>  The centered octagonal numbers are the same as the [[odd number|odd]] [[square number]]s.<ref name="oeis"/> Thus, the ''n''th odd square number and ''t''th centered octagonal number is given by the formula
:<math>O_n=(2n-1)^2 = 4n^2-4n+1 | (2t+1)^2=4t^2+4t+1.</math>

[[Image:visual_proof_centered_octagonal_numbers_are_odd_squares.svg|thumb|upright|[[Proof without words]] that all centered octagonal numbers are odd squares]]
The first few centered octagonal numbers are<ref name="oeis">{{Cite OEIS|A016754|name=Odd squares: (2n-1)^2. Also centered octagonal numbers.}}</ref>
:[[1 (number)|1]], [[9 (number)|9]], [[25 (number)|25]], [[49 (number)|49]], [[81 (number)|81]], [[121 (number)|121]], [[169 (number)|169]], 225, 289, 361, 441, 529, 625, 729, 841, 961, [[1089 (number)|1089]], 1225

Calculating [[Ramanujan's tau function]] on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.<ref name="oeis"/>

<math>O_n</math> is the number of 2x2 matrices with elements from 0 to n that their [[determinant]] is twice their [[Permanent (mathematics)|permanent]].

==See also==
* [[Octagonal number]]

==References==
{{reflist}}

{{Figurate numbers}}
{{Classes of natural numbers}}
{{DEFAULTSORT:Centered Octagonal Number}}
[[Category:Figurate numbers]]