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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 n-th octagonal number is
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.
Sum of reciprocals
Test for octagonal numbers
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.
- Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, p. 57, ISBN 9789814355483.
- Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers