Tetrahedral Number as Sum of Squares/Examples

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Examples of Tetrahedral Number as Sum of Squares

Let $H_n$ denote the $n$th tetrahedral number.

Then:

\(\ds H_1\) \(=\) \(\, \ds 1 \, \) \(\, \ds = \, \) \(\ds 1^2\)
\(\ds H_2\) \(=\) \(\, \ds 4 \, \) \(\, \ds = \, \) \(\ds 2^2\)
\(\ds H_3\) \(=\) \(\, \ds 10 \, \) \(\, \ds = \, \) \(\ds 1^2 + 3^2\)
\(\ds H_4\) \(=\) \(\, \ds 20 \, \) \(\, \ds = \, \) \(\ds 2^2 + 4^2\)
\(\ds H_5\) \(=\) \(\, \ds 35 \, \) \(\, \ds = \, \) \(\ds 1^2 + 3^2 + 5^2\)
\(\ds H_6\) \(=\) \(\, \ds 56 \, \) \(\, \ds = \, \) \(\ds 2^2 + 4^2 + 6^2\)
\(\ds H_7\) \(=\) \(\, \ds 84 \, \) \(\, \ds = \, \) \(\ds 1^2 + 3^2 + 5^2 + 7^2\)

... and so on.


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