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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $8$