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}} 200px|right

A '''centered octagonal number''' is a 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 square numbers.<ref name="oeis"/> Thus, the ''n''th odd square number and ''t''th centered octagonal number is given by :<math>O_n = (2n-1)^2 = 4n^2-4n+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, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 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 <math>2 \times 2</math> matrices with elements from <math>0</math> to <math>n</math> whose determinant and permanent are both zero, i.e. that have an either a row or column that is identically zero.

==See also== * Octagonal number

==References== {{reflist}}

{{DEFAULTSORT:Centered Octagonal Number}} Category:Figurate numbers