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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
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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]
The th octagonal number is the number of partitions of into 1, 2, or 3s.[2] For example, there are such partitions for , namely
A formula for the sum of the reciprocals of the octagonal numbers is given by[3]
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.
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