Acentered octagonal number is acenteredfigurate number that represents anoctagon 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 theoddsquare numbers.^[2] Thus, thenth odd square number andtth centered octagonal number is given by the formula

${\displaystyle O_n=(2n-1{)}^2=4n^2-4n+1|(2t+1{)}^2=4t^2+4t+1.}$

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

CalculatingRamanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.^[2]

${\displaystyle O_n}$ is the number of${\displaystyle 2\times 2}$ matrices with elements from${\displaystyle 0}$ to${\displaystyle n}$ whosedeterminant andpermanent are both zero, i.e. that have a either a row or column that is identically zero.

• Octagonal number

• Figurate numbers

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