Centered octagonal number explained

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. Thus, the nth odd square number and tth centered octagonal number is given by the formula

O ^{2} _{n = ( 2 n -1 )} = 4 n ^{2-4n+1} | ( 2 t +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]

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

See also

• Octagonal number

Notes and References

1. .
2. Odd squares: (2n-1)^2. Also centered octagonal numbers..