A Tiling Approach to Chebyshev Polynomials

We present a combinatorial interpretation of Chebyshev polynomials. The nth Chebyshev polynomial of the first kind, Tn(x), counts the sum of all weights of n-tilings using light and dark squares of weight x and dominoes of weight −1, and the first tile, if a square must be light. If we relax the condition that the first square must be light, the sum of all weights is the nth Chebyshev polynomial of the second kind, Un(x). In this paper we prove many of the beautiful Chebyshev identities using the tiling interpretation