Triangular Lucas Numbers

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

$\ds 1$

$=$

$\ds \dfrac {1 \times 2} 2$

$\ds 3$

$=$

$\ds \dfrac {2 \times 3} 2$

$\ds = 2 + 1$

$\ds 5778$

$=$

$\ds \dfrac {107 \times 108} 2$

$\ds = 2207 + 3571$

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Sources

1989: Luo Ming: On Triangular Fibonacci Numbers (The Fibonacci Quarterly Vol. 27, no. 2: pp. 98 – 108)

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5778$