Triangular Lucas Numbers

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Theorem

The only Lucas numbers which are also triangular are:

$1, 3, 5778$

This sequence is A248506 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle 1\) \(=\) \(\displaystyle \dfrac {1 \times 2} 2\)
\(\displaystyle 3\) \(=\) \(\displaystyle \dfrac {2 \times 3} 2\) \(\displaystyle = 2 + 1\)
\(\displaystyle 5778\) \(=\) \(\displaystyle \dfrac {107 \times 108} 2\) \(\displaystyle = 2207 + 3571\)



Sources