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. The centered octagonal number for n is given by the formula
where T is a regular triangular number, or much more simply, by squaring the odd numbers:
The first few centered octagonal numbers are
1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089
All centered octagonal numbers are odd, and in base 10 one can notice the one's digits follow the pattern 1-9-5-9-1. An odd number is a centered octagonal number if and only if it is a perfect square.
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.
See also regular octagonal number.
Famous quotes containing the words centered and/or number:
“The difference between style and taste is never easy to define, but style tends to be centered on the social, and taste upon the individual. Style then works along axes of similarity to identify group membership, to relate to the social order; taste works within style to differentiate and construct the individual. Style speaks about social factors such as class, age, and other more flexible, less definable social formations; taste talks of the individual inflection of the social.”
—John Fiske (b. 1939)
“If we remembered everything, we should on most occasions be as ill off as if we remembered nothing. It would take us as long to recall a space of time as it took the original time to elapse, and we should never get ahead with our thinking. All recollected times undergo, accordingly, what M. Ribot calls foreshortening; and this foreshortening is due to the omission of an enormous number of facts which filled them.”
—William James (1842–1910)