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 The first few centered octagonal numbers are[2] 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]

is the number of 2x2 matrices with elements from 0 to n that their determinant is twice their permanent.