octagonal number

{{Short description|Number of points in an octagonal arrangement}}

{{more citations needed|date=October 2013}}

File:OctagonalNumbers.svg

In mathematics, an octagonal number is a figurate number. The nth octagonal number on 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 overlaid so that they share one vertex. The octagonal number for n is given by the formula 3n2 − 2n, with n > 0. The first few octagonal numbers are

: 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936 {{OEIS|id=A000567}}

The octagonal number for n can also be calculated by adding the square of n to twice the (n − 1)th pronic number.

Octagonal numbers consistently alternate parity.

Octagonal numbers are occasionally referred to as "star numbers", though that term is more commonly used to refer to centered dodecagonal numbers.{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|author1-link=Elena Deza|first2=Michel|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=57|url=https://books.google.com/books?id=cDxYdstLPz4C&pg=PA57}}.

Applications in combinatorics

The nth octagonal number is the number of partitions of 6n-5 into 1, 2, or 3s.{{OEIS|id=A000567}} For example, there are x_2=8 such partitions for 2\cdot 6-5=7, namely

: [1,1,1,1,1,1,1], [1,1,1,1,1,2], [1,1,1,1,3], [1,1,1,2,2], [1,1,2,3], [1,2,2,2], [1,3,3] and [2,2,3].

Sum of reciprocals

A formula for the sum of the reciprocals of the octagonal numbers is given by{{Cite web |url=http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |title=Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers |access-date=2020-04-12 |archive-date=2013-05-29 |archive-url=https://web.archive.org/web/20130529032918/http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |url-status=dead }}

\sum_{n=1}^\infty \frac{1}{n(3n-2)} = \frac{9\ln(3)+\sqrt3\pi}{12}.

Test for octagonal numbers

Solving the formula for the n-th octagonal number, x_n, for n gives

n= \frac{\sqrt{3x_n+1}+1}{3}.

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

References

{{reflist}}

{{Figurate numbers}}

{{Classes of natural numbers}}

Category:Figurate numbers

{{num-stub}}