A centered octagonal number is a
centered figurate number 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, wh ...
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 of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The u ...
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
* 121 (MBTA bus)
* 121 (New Jersey bus)
*Road 121, see list of highways numbered 121
*Russian cruiser ...
,
169
Year 169 ( CLXIX) was a common year starting on Saturday (link will display the full calendar) 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 co ...
, 225, 289, 361, 441, 529, 625, 729, 841, 961,
1089
Year 1089 ( MLXXXIX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* King Demetrius Zvonimir of Croatia dies after a 12-year reign, and is succeeded by Step ...
, 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.
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 ...
is twice their
permanent.
See also
*
Octagonal number
References
{{DEFAULTSORT:Centered Octagonal Number
Figurate numbers