Centered octagonal number

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. ^[1] The centered octagonal numbers are the same as the odd square numbers. ^[2] Thus, the nth odd square number and tth centered octagonal number is given by the formula

${\displaystyle O_{n}=(2n-1)^{2}=4n^{2}-4n+1|(2t+1)^{2}=4t^{2}+4t+1.}$

Proof without words that all centered octagonal numbers are odd squares

The first few centered octagonal numbers are ^[2]

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. ^[2]

${\displaystyle O_{n}}$ is the number of ${\displaystyle 2\times 2}$ matrices with elements from ${\displaystyle 0}$ to ${\displaystyle n}$ whose determinant and permanent are both zero, i.e. that have a either a row or column that is identically zero.

See also

• Octagonal number

References

1. Teo, Boon K.; Sloane, N. J. A. (1985), "Magic numbers in polygonal and polyhedral clusters" (PDF), Inorganic Chemistry, 24 (26): 4545–4558, doi:10.1021/ic00220a025 .
2. Sloane, N. J. A. (ed.). "SequenceA016754(Odd squares: (2n-1)^2. Also centered octagonal numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.