Tetrahedral Number as Sum of Squares
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Theorem
- $H_n = \ds \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2$
where $H_n$ denotes the $n$th tetrahedral number.
Examples
\(\ds H_1\) | \(=\) | \(\, \ds 1 \, \) | \(\, \ds = \, \) | \(\ds 1^2\) | ||||||||||
\(\ds H_2\) | \(=\) | \(\, \ds 4 \, \) | \(\, \ds = \, \) | \(\ds 2^2\) | ||||||||||
\(\ds H_3\) | \(=\) | \(\, \ds 10 \, \) | \(\, \ds = \, \) | \(\ds 1^2 + 3^2\) | ||||||||||
\(\ds H_4\) | \(=\) | \(\, \ds 20 \, \) | \(\, \ds = \, \) | \(\ds 2^2 + 4^2\) | ||||||||||
\(\ds H_5\) | \(=\) | \(\, \ds 35 \, \) | \(\, \ds = \, \) | \(\ds 1^2 + 3^2 + 5^2\) | ||||||||||
\(\ds H_6\) | \(=\) | \(\, \ds 56 \, \) | \(\, \ds = \, \) | \(\ds 2^2 + 4^2 + 6^2\) | ||||||||||
\(\ds H_7\) | \(=\) | \(\, \ds 84 \, \) | \(\, \ds = \, \) | \(\ds 1^2 + 3^2 + 5^2 + 7^2\) |
... and so on.
Proof
Let $n$ be even such that $n = 2 m$.
We have:
\(\ds \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2\) | \(=\) | \(\ds \sum_{k \mathop = 0}^m \paren {2 m - 2 k}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^m \paren {2 k}^2\) | Permutation of Indices of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 m \paren {m + 1} \paren {2 m + 1} } 3\) | Sum of Sequence of Even Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 m \paren {2 m + 1} \paren {2 m + 2} } 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H_{2 m}\) | Closed Form for Tetrahedral Numbers |
Let $n$ be odd such that $n = 2 m + 1$.
We have:
\(\ds \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2\) | \(=\) | \(\ds \sum_{k \mathop = 0}^m \paren {2 m + 1 - 2 k}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^m \paren {2 k + 1}^2\) | Permutation of Indices of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {m + 1} \paren {2 m + 1} \paren {2 m + 3} } 3\) | Sum of Sequence of Odd Squares: Formulation 1 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {2 m + 1} \paren {2 m + 2} \paren {2 m + 3} } 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H_{2 m + 1}\) | Closed Form for Tetrahedral Numbers |
$\blacksquare$