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 $n\times1$) with squares of unit width and $(1,1)$-fence tiles. A $(1,1)$-fence tile is composed of two unit-width square subtiles 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 consecutive 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 identities involving sums of Fibonacci and Jacobsthal numbers in a straightforward way. Some of these identities appear to be new. We also construct and obtain identities for a known Pascal-like triangle (which has 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 analogous one for tilings of an $n$-board. Connections between the triangles and Riordan arrays are also demonstrated. With the help of the triangles, we express the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers as double sums of products of two binomial coefficients.