Identities involving the tribonacci numbers squared via tiling with combs

TL;DR

This paper introduces a tiling method using combs to establish identities involving squared tribonacci numbers and connects them to Fibonacci, Narayana's cows, and Padovan numbers, providing new combinatorial proofs.

Contribution

It presents a novel tiling approach with combs to derive and prove identities involving tribonacci numbers and related sequences.

Findings

Derived identities relating tribonacci numbers squared to other sequences

Established combinatorial proofs for these identities

Most identities are newly discovered

Abstract

The number of ways to tile an $n$-board (an $n\times1$ rectangular board) with $(\frac12,\frac12;1)$-, $(\frac12,\frac12;2)$-, and $(\frac12,\frac12;3)$-combs is $T_{n+2}^2$ where $T_n$ is the $n$th tribonacci number. A $(\frac12,\frac12;m)$-comb is a tile composed of $m$ sub-tiles of dimensions $\frac12\times1$ (with the shorter sides always horizontal) separated by gaps of dimensions $\frac12\times1$. We use such tilings to obtain quick combinatorial proofs of identities relating the tribonacci numbers squared to one another, to other combinations of tribonacci numbers, and to the Fibonacci, Narayana's cows, and Padovan numbers. Most of these identities appear to be new.