Chebyshev polynomials have several elegant combinatorial interpretations. Specificially, the Chebyshev polynomials of the first kind are defined by T0(x) = 1, T1(x) = x, and Tn(x) = 2x Tn-1(x) - Tn-2(x). Chebyshev polynomials of the second kind Un(x) are defined the same way, except U1(x) = 2x. Tn and Un are shown to count tilings of length n strips with squares and dominoes, where the tiles are given weights and sometimes color. Using these interpretations, many identities satisfied by Chebyshev polynomials can be given combinatorial proofs.
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Benjamin, A.T., & Walton, D. (2009). Counting on Chebyshev Polynomials. Mathematics Magazine, 82(2): 117-126.