Closed Form for Heptagonal Pyramidal Numbers

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Theorem

The closed-form expression for the $n$th heptagonal pyramidal number is:

$Q_n = \dfrac {n \paren {n + 1} \paren {5 n - 2} } 6$


Proof

\(\ds Q_n\) \(=\) \(\ds \sum_{k \mathop = 1}^n H_n\) Definition of Heptagonal Pyramidal Number
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \dfrac {k \paren {5 k - 3} } 2\) Closed Form for Heptagonal Numbers
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {5 \sum_{k \mathop = 1}^n k^2 - 3 \sum_{k \mathop = 1}^n k}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {5 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \sum_{k \mathop = 1}^n k}\) Sum of Sequence of Squares
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {5 \frac {n \paren {n + 1} \paren {2 n + 1} } 6 - 3 \dfrac {n \paren {n + 1} } 2}\) Closed Form for Triangular Numbers
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\frac {5 n \paren {n + 1} \paren {2 n + 1} - 9 n \paren {n + 1} } 6}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {5 \paren {2 n + 1} - 9} } 6}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {10 n + 5 - 9} } 6}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\frac {n \paren {n + 1} \paren {10 n - 4} } 6}\)
\(\ds \) \(=\) \(\ds \frac {n \paren {n + 1} \paren {5 n - 2} } 6\)

$\blacksquare$


Sources

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