Chebyshev Polynomials, Sliding Columns, and the k-Step Fibonacci Numbers

Abstract

Summary We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the Fibonaccis, the Tribonaccis, the Tetranaccis, and so on.