Octagonal number

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An octagonal number is a figurate number that represents an octagon. 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)

Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the n-th octagonal number is

${\displaystyle x_{n}=n^{2}+4\sum _{k=1}^{n-1}k=3n^{2}-2n.}$

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]

Sum of reciprocals[]

A formula for the sum of the reciprocals of the octagonal numbers is given by:^[2]

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(3n-2)}}={\frac {9\ln(3)+{\sqrt {3}}\pi }{12}}}$

Test for octagonal numbers[]

Solving the formula for the n-th octagonal number, ${\displaystyle x_{n},}$ for n gives

${\displaystyle n={\frac {{\sqrt {3x_{n}+1}}+1}{3}}.}$

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[]

• Centered octagonal number

References[]

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