An Interesting Relationship Between Squares and Hex Numbers

FormalPara Proposition.

Let h_nstand for the nth hexagonal number. Then for$n \in {\mathbb{N}}$, the following identity holds:

$$ h_{3n - 2} = 6h_{n} + 1h_{n - 1} + 6\left( {n - 1} \right)^{2} - 6n $$

FormalPara Proof.

The proof is demonstrated for n = 5.□