Closed Form for Centered Hexagonal Numbers
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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): $37$