Closed Form for Hexagonal Pyramidal Numbers
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Theorem
The closed-form expression for the $n$th hexagonal pyramidal number is:
- $S_n = \dfrac {n \paren {n + 1} \paren {4 n - 1} } 6$
Proof
| \(\ds S_n\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n H_k\) | Definition of Hexagonal Pyramidal Number | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n k \paren {2 k - 1}\) | Closed Form for Hexagonal Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \sum_{k \mathop = 1}^n 2 k^2 - \sum_{k \mathop = 1}^n k\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} } 6 - \sum_{k \mathop = 1}^n k\) | Sum of Sequence of Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} } 6 - \dfrac {n \paren {n + 1} } 2\) | Closed Form for Triangular Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 n \paren {n + 1} \paren {2 n + 1} - 3 n \paren {n + 1} } 6\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n \paren {n + 1} \paren {4 n + 2 - 3} } 6\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n \paren {n + 1} \paren {4 n - 1} } 6\) |
$\blacksquare$