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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
which simplifies to the formulas 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, .
Octagonal numbers consistently alternate parity.
Test for octagonal numbers
If x is an integer, then n is the xth octagonal number. If x is not an integer, then n is not octagonal.
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