Octagonal number
An octagonal number is a figurate number that gives the number of points in a certain octagonal arrangement. 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.[1]
Applications in combinatorics
The th octagonal number is the number of partitions of into 1, 2, or 3s.[2] For example, there are such partitions for , 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[3]
Test for octagonal numbers
Solving the formula for the n-th octagonal number, for n gives
See also
- Centered octagonal number
References
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