Octagonal number
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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 (sequence A000567 in the OEIS)
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.<ref>Template:Citation.</ref>
Applications in combinatorics
The <math>n</math>th octagonal number is the number of partitions of <math>6n-5</math> into 1, 2, or 3s.<ref>(sequence A000567 in the OEIS)</ref> For example, there are <math>x_2=8</math> such partitions for <math>2\cdot 6-5=7</math>, 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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display=block> \sum_{n=1}^\infty \frac{1}{n(3n-2)} = \frac{9\ln(3)+\sqrt3\pi}{12}. </math>
Test for octagonal numbers
Solving the formula for the n-th octagonal number, <math>x_n,</math> for n gives <math display=block>n= \frac{\sqrt{3x_n+1}+1}{3}.</math> 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
Template:Figurate numbers Template:Classes of natural numbers