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

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

The first few centered octagonal numbers are²

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 an either a row or column that is identically zero.