Closed Form for Centered Hexagonal Numbers

Theorem

Let $C_n$ be the $n$th centered hexagonal number.

Then:

$C_n = 3 n \paren {n - 1} + 1$

Proof

By the definition of centered hexagonal number:

\(\ds C_n\)

\(=\)

\(\ds 1 + \sum_{k \mathop = 1}^{n - 1} 6 k\)

Definition of Centered Hexagonal Number

\(\ds \)

\(=\)

\(\ds 1 + 6 \sum_{k \mathop = 1}^{n - 1} k\)

\(\ds \)

\(=\)

\(\ds 6 \paren {\dfrac {n \paren {n - 1} } 2} + 1\)

Closed Form for Triangular Numbers

\(\ds \)

\(=\)

\(\ds 3 n \paren {n - 1} + 1\)

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $37$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $37$