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

\(\ds 1\)

\(=\)

\(\ds \dfrac {1 \times 2} 2\)

\(\ds 3\)

\(=\)

\(\ds \dfrac {2 \times 3} 2\)

\(\ds = 2 + 1\)

\(\ds 5778\)

\(=\)

\(\ds \dfrac {107 \times 108} 2\)

\(\ds = 2207 + 3571\)

This theorem requires a proof. In particular: It remains to be shown that these are the only ones. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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): $5778$