Triangular Numbers which are also Pentagonal

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Theorem

The sequence of triangular numbers which are also pentagonal begins:

$1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$

This sequence is A014979 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\ds 1\) \(=\) \(\ds \dfrac {1 \paren {3 \times 1 - 1} } 2\) Closed Form for Pentagonal Numbers
\(\ds \) \(=\) \(\ds \dfrac {1 \times \paren {1 + 1} } 2\) Closed Form for Triangular Numbers


\(\ds 210\) \(=\) \(\ds \dfrac {12 \paren {3 \times 12 - 1} } 2\) Closed Form for Pentagonal Numbers
\(\ds \) \(=\) \(\ds \dfrac {20 \times \paren {20 + 1} } 2\) Closed Form for Triangular Numbers


\(\ds 40 \, 755\) \(=\) \(\ds \dfrac {165 \paren {3 \times 165 - 1} } 2\) Closed Form for Pentagonal Numbers
\(\ds \) \(=\) \(\ds \dfrac {285 \times \paren {285 + 1} } 2\) Closed Form for Triangular Numbers



Sources