Octagonal 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 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.

, 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,

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.

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