Domino Tilings of a Checkerboard

for n ≥ 2, it can be shown that fn counts the number of ways to tile a 1 × n board with squares and dominoes. This interpretation of the Fibonacci numbers admits clever counting proofs of many Fibonacci identities. When we consider more complex combinatorial objects, we find that simply counting the number of tilings is not quite enough, and that we must consider instead weighted tilings. For example, Chebyshev polynomials of the second kind count the total weights of all tilings of a 1 × n board, where squares are given a weight of 2x and dominoes assigned a weight of −1, and the weight of a tiling is simply the product of the weights of the tiles used [1]. This interpretation of the Chebyshev polynomials can be used to prove, for example,