IDENTITIES INVOLVING THE TRIBONACCI NUMBERS SQUARED VIA TILING WITH COMBS - Allen M.A. (2023)
The number of ways to tile an n-board (an n×1 rectangular board) with (2112 ; 1)-, (2112 ; 2)-, and (2112 ; 3)-combs is Tn2+2, where Tn is the nth tribonacci number. A (2112 ; m)comb is a tile composed of m sub-tiles of dimensions 12 × 1 (with the shorter sides always horizontal) separated by gaps of dimensions 21 × 1. 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.