Closed Form for Octagonal Numbers

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Theorem

The closed-form expression for the $n$th octagonal number is:

$O_n = n \paren {3 n - 2}$


Proof

Octagonal numbers are $k$-gonal numbers where $k = 8$.

From Closed Form for Polygonal Numbers we have that:

$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$


Hence:

\(\ds O_n\) \(=\) \(\ds \frac n 2 \paren {\paren {8 - 2} n - 8 + 4}\) Closed Form for Polygonal Numbers
\(\ds \) \(=\) \(\ds n \paren {3 n - 2}\)

Hence the result.

$\blacksquare$


Sources