Centered octagonal number
Centered figurate number that represents an octagon with a dot in the center
title: "Centered octagonal number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["figurate-numbers"] description: "Centered figurate number that represents an octagon with a dot in the center" topic_path: "general/figurate-numbers" source: "https://en.wikipedia.org/wiki/Centered_octagonal_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Centered figurate number that represents an octagon with a dot in the center ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/6c/Centered_octagonal_number.svg"] ::
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.{{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}}. The centered octagonal numbers are the same as the odd square numbers. Thus, the nth odd square number and tth centered octagonal number is given by the formula :O_n=(2n-1)^2 = 4n^2-4n+1 | (2t+1)^2=4t^2+4t+1.
::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/0e/visual_proof_centered_octagonal_numbers_are_odd_squares.svg" caption="[[Proof without words]] that all centered octagonal numbers are odd squares"] ::
The first few centered octagonal numbers are :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.
O_n is the number of 2 \times 2 matrices with elements from 0 to n whose determinant and permanent are both zero, i.e. that have a either a row or column that is identically zero.
References
References
- {{Cite OEIS. A016754
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