centered octagonal number

{{Short description|Centered figurate number that represents an octagon with a dot in the center}}

{{Use American English|date=March 2021}}

{{Use mdy dates|date=March 2021}}

Image: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.{{citation

| last1 = Teo | first1 = Boon K.

| last2 = Sloane | first2 = N. J. A. | author2-link = Neil Sloane

| journal = Inorganic Chemistry

| pages = 4545–4558

| title = Magic numbers in polygonal and polyhedral clusters

| url = http://neilsloane.com/doc/magic1/magic1.pdf

| volume = 24

| issue = 26

| year = 1985 | doi=10.1021/ic00220a025}}. The centered octagonal numbers are the same as the odd square numbers. Thus, the nth odd square number and tth centered octagonal number is given by the formula

:$O\_n=(2n-1)^2\; =\; 4n^2-4n+1\; |\; (2t+1)^2=4t^2+4t+1.$

Image:visual_proof_centered_octagonal_numbers_are_odd_squares.svg that all centered octagonal numbers are odd squares]]

The first few centered octagonal numbers are{{Cite OEIS|A016754|name=Odd squares: (2n-1)^2. Also centered octagonal numbers.}}

: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.

$O\_n$ is the number of 2x2 matrices with elements from 0 to n that their determinant is twice their permanent.