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]

See also

• Octagonal number

References

1. Teo, Boon K.; Sloane, N. J. A. (1985), "Magic numbers in polygonal and polyhedral clusters" (PDF), Inorganic Chemistry 24: 4545–4558, doi:10.1021/ic00220a025.
2. "Sloane's A016754 : Odd squares: (2n+1)^2. Also centered octagonal numbers.", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.