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

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

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]