Closed Form for Hexagonal Numbers
Theorem
The closed-form expression for the $n$th hexagonal number is:
- $H_n = n \paren {2 n - 1}$
Proof
Hexagonal numbers are $k$-gonal numbers where $k = 6$.
From Closed Form for Polygonal Numbers we have that:
- $\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
| \(\ds H_n\) | \(=\) | \(\ds \frac n 2 \paren {\paren {6 - 2} n - 6 + 4}\) | Closed Form for Polygonal Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds n \paren {2 n - 1}\) |
Hence the result.
$\blacksquare$