# A new combinatorial interpretation of the Fibonacci numbers squared

## TL;DR

This paper introduces a novel combinatorial interpretation of Fibonacci numbers squared through tiling problems involving half-squares and fence tiles, providing new proofs and identities related to Fibonacci numbers.

## Contribution

It presents a new tiling-based combinatorial interpretation of Fibonacci numbers squared and derives related identities, some of which are newly discovered.

## Key findings

- Number of tilings equals Fibonacci number squared.
- Derived new identities relating Fibonacci numbers squared.
- Provided combinatorial proofs for these identities.

## Abstract

We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares separated by a gap of width $\frac12$. We show that the number of ways to tile an $n$-board using these types of tiles equals $F_{n+1}^2$ where $F_n$ is the $n$th Fibonacci number. We use these tilings to devise combinatorial proofs of identities relating the Fibonacci numbers squared to one another and to other number sequences. Some of these identities appear to be new.