A centered octagonal number is a
centered figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polygon ...
that represents an
octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.
[.] The centered octagonal numbers are the same as the
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s.
Thus, the ''n''th odd square number and ''t''th centered octagonal number is given by the formula
:
The first few centered octagonal numbers are
:
1,
9,
25,
49,
81,
121,
169, 225, 289, 361, 441, 529, 625, 729, 841, 961,
1089, 1225
Calculating
Ramanujan's tau function
The Ramanujan tau function, studied by , is the function \tau : \mathbb \rarr\mathbb defined by the following identity:
:\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z),
where with , \phi is the Euler function, is the ...
on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.
is the number of 2x2 matrices with elements from 0 to n that their
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
is twice their
permanent
Permanent may refer to:
Art and entertainment
* ''Permanent'' (film), a 2017 American film
* ''Permanent'' (Joy Division album)
* "Permanent" (song), by David Cook
Other uses
* Permanent (mathematics), a concept in linear algebra
* Permanent (cy ...
.
See also
*
Octagonal number
An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are
: 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 34 ...
References
{{DEFAULTSORT:Centered Octagonal Number
Figurate numbers