Centered octagonal number
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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 number for n is given by the formula
where T is a regular triangular number, or much more simply, by squaring the odd numbers:
The first few centered octagonal numbers are
All centered octagonal numbers are odd, and in base 10 one can notice the one's digits follow the pattern 1-9-5-9-1. An odd number is a centered octagonal number if and only if it is a perfect square.
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.
See also regular octagonal number.