Centered octagonal number

File:Centered octagonal number.svg

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.<ref>Template:Citation.</ref> The centered octagonal numbers are the same as the odd square numbers.<ref name="oeis"/> Thus, the nth odd square number and tth centered octagonal number is given by the formula

<math>O_n=(2n-1)^2 = 4n^2-4n+1 | (2t+1)^2=4t^2+4t+1.</math>

File:Visual proof centered octagonal numbers are odd squares.svg

Proof without words that all centered octagonal numbers are odd squares

The first few centered octagonal numbers are<ref name="oeis">Template:Cite OEIS</ref>

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.<ref name="oeis"/>

<math>O_n</math> is the number of <math>2 \times 2</math> matrices with elements from <math>0</math> to <math>n</math> whose determinant and permanent are both zero, i.e. that have a either a row or column that is identically zero.

See also

• Octagonal number