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.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $45$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $45$