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.{{citation :O_n=(2n-1)^2 = 4n^2-4n+1 | (2t+1)^2=4t^2+4t+1.

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.