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. 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

The first few centered octagonal numbers are

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.

${\displaystyle O_{n}}$ 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.