Square of Odd Multiple of 3 is Difference between Triangular Numbers

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Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $T_n$ denote the $n$th triangular number.

Let $m = 2 n + 1$ be an odd integer


Then:

$\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$


Proof

\(\ds T_{9 n + 4} - T_{3 n + 1}\) \(=\) \(\ds \dfrac {\paren {9 n + 4} \paren {9 n + 5} } 2 - \dfrac {\paren {3 n + 1} \paren {3 n + 2} } 2\) Closed Form for Triangular Numbers
\(\ds \) \(=\) \(\ds \dfrac {\paren {81 n^2 + 81 n + 20} - \paren {9 n^2 + 9 n + 2} } 2\)
\(\ds \) \(=\) \(\ds \dfrac {72 n^2 + 72 n + 18} 2\)
\(\ds \) \(=\) \(\ds 36 n^2 + 36 n + 9\)
\(\ds \) \(=\) \(\ds 9 \paren {4 n^2 + 4 n + 1}\)
\(\ds \) \(=\) \(\ds 9 \paren {2 n + 1}^2\)
\(\ds \) \(=\) \(\ds 9 m^2\)
\(\ds \) \(=\) \(\ds \paren {3 m}^2\)

$\blacksquare$


Sources