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 ancient Greek mathemat ...
that represents an
octagon
In geometry, an octagon () 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, which alternates two types of edges. A truncated octagon, t is a ...
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 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,
121 121 may refer to:
*121 (number), a natural number
* AD 121, a year in the 2nd century AD
* 121 BC, a year in the 2nd century BC
* 121 (Eagle) Sqn, a Royal Air Force aircraft squadron that during the Second World War was one of the three Eagle Squa ...
,
169
Year 169 (Roman numerals, CLXIX) was a common year starting on Saturday of the Julian calendar. At the time, it was known as the Year of the Consulship of Senecio and Apollinaris (or, less frequently, year 922 ''Ab urbe condita''). The denominat ...
, 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\to\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 q=\exp(2\pi iz) with \mathrm(z)>0, \phi is t ...
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 (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
is twice their
permanent.
See also
*
Octagonal number
In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o'n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlai ...
References
{{DEFAULTSORT:Centered Octagonal Number
Figurate numbers