New combinatorial interpretations of the fibonacci numbers squared, golden rectangle numbers, and jacobsthal numbers using two types of tile

Abstract

We consider the tiling of an n-board (a board of size n × 1) with squares of unit width and (1, 1)-fence tiles. A (1, 1)-fence tile is composed of two unit-width square sub-tiles separated by a gap of unit width. We show that the number of ways to tile an n-board using unit-width squares and (1, 1)-fence tiles is equal to a Fibonacci number squared when n is even and a golden rectangle number (the product of two consec-utive Fibonacci numbers) when n is odd. We also show that the number of tilings of boards using n such square and fence tiles is a Jacobsthal number. Using combinatorial techniques we prove new identities involving golden rectangle and Jacobsthal numbers. Two of the identities involve entries in two Pascal-like triangles. One is a known triangle (with alternating ones and zeros along one side) whose (n, k)th entry is the number of tilings using n tiles of which k are fence tiles. There is a simple relation between this triangle and the other which is the analogous triangle for tilings of an n-board. These triangles are related to Riordan arrays and we give a general procedure for finding which Riordan array(s) a triangle is related to. The resulting combinatorial interpretation of the Riordan arrays allows one to derive properties of them via combinatorial proof.