An octagonal number is a
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 ...
. 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,
133 133 may refer to:
*133 (number)
* AD 133
*133 BC
*133 (song)
*133 (New Jersey bus) 133 may refer to:
*133 (number)
* AD 133
*133 BC
*133 (song) 133 may refer to:
*133 (number)
*AD 133
*133 BC
*133 (song)
*133 (New Jersey bus) 133 may refer to:
* ...
,
176
Year 176 ( CLXXVI) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Proculus and Aper (or, less frequently, year 929 '' Ab urbe condita'') ...
, 225,
280
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Year 280 ( CCLXXX) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Messalla and Gratus (or, less frequently, year 1033 '' ...
, 341, 408, 481, 560, 645, 736, 833, 936
Octagonal numbers can be formed by placing
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s on the four sides of a square. To put it algebraically, the ''n''-th octagonal number is
The octagonal number for ''n'' can also be calculated by adding the square of ''n'' to twice the (''n - 1'')th
pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...
.
Octagonal numbers consistently alternate
parity.
Octagonal numbers are occasionally referred to as "
star number
A star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on.
The ''n''th star number is given by the formula ''Sn'' = 6''n''(''n'' − 1) + 1. The ...
s," though that term is more commonly used to refer to centered dodecagonal numbers.
Applications in combinatorics
is the number of partitions of
into 1,2 or 3s. For example: there are
such partitions for
:
:
,1,1,1,1,1,1 ,1,1,1,1,2 ,1,1,1,3 ,1,1,2,2 ,1,2,3 ,2,2,2 ,3,3and
,2,3
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
Sum of reciprocals
A formula for the
sum of the reciprocals of the octagonal numbers is given by
Test for octagonal numbers
Solving the formula for the ''n''-th octagonal number,
for ''n'' gives
An arbitrary number ''x'' can be checked for octagonality by putting it in this equation. If ''n'' is an integer, then ''x'' is the ''n''-th octagonal number. If ''n'' is not an integer, then ''x'' is not octagonal.
See also
*
Centered octagonal number
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 numbers are the same as the od ...
References
Figurate numbers
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