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We consider the tiling of an $n$-board (a board of size $n\times1$) withsquares of unit width and $(1,1)$-fence tiles. A $(1,1)$-fence tile is composedof two unit-width square subtiles separated by a gap of unit width. We showthat 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 anda 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 combinatorialtechniques we prove identities involving sums of Fibonacci and Jacobsthalnumbers in a straightforward way. Some of these identities appear to be new. Wealso construct and obtain identities for a known Pascal-like triangle (whichhas alternating ones and zeros along one side) whose $(n,k)$th entry is thenumber of tilings using $n$ tiles of which $k$ are fence tiles. There is asimple relation between this triangle and the analogous one for tilings of an$n$-board. Connections between the triangles and Riordan arrays are alsodemonstrated. With the help of the triangles, we express the Fibonacci numberssquared, golden rectangle numbers, and Jacobsthal numbers as double sums ofproducts of two binomial coefficients.

New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile