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): $37$
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