A new combinatorial interpretation of the fibonacci numbers squared. Part ii.

Abstract

© 2020 Fibonacci Association. All rights reserved. We give further combinatorial proofs of identities related to the Fibonacci num-bers squared by considering the tiling of an n-board (a 1 ×n array of square cells of unit width) with half-squares ( 1/2 ×1 tiles) and (1/2,1/2 )-fence tiles. A (w; g)-fence tile is composed of two w × 1 rectangular subtiles separated by a gap of width g. In addition, we construct a Pascal-like triangle whose (n; k)th entry is the number of tilings of an n-board that contain k fences. Elementary combinatorial proofs are given for some properties of the triangle and we show that reversing the rows gives the (1/(1-x2); x/(1-x)2) Riordan array. Finally, we show that tiling an n-board with (1/4 1/4 )-and (1/4 ; 3/4 )-fences also generates the Fibonacci numbers squared.