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Centered octagonal number

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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 centered octagonal number is given by the formula

(2n-1)^2 = 4n^2-4n+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.

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]

References

1. ^ Teo, Boon K.; .
2. ^ a b c "Sloane's A016754 : Odd squares: (2n+1)^2. Also centered octagonal numbers.", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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