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 OEIS)

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

$n^2 + 4\sum_{k = 1}^{n - 1} k$

which simplifies to the formula given above.

The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number, or, to put it algebraically, $O n = n 2 + 2(n 2 - n)$ .

Octagonal numbers consistently alternate parity.

Confusingly, octagonal numbers are occasionally referred to as "star numbers," though that term is more commonly used to refer to centered dodecagonal numbers.