Chebyshev Polynomials, Sliding Columns, and the $k$-step Fibonacci Numbers

TL;DR

This paper presents an intuitive proof connecting Chebyshev polynomials to Fibonacci-like sequences through a visual column sliding method, revealing new insights into their combinatorial relationships.

Contribution

It introduces a novel, visual proof technique linking Chebyshev polynomials with Fibonacci and related sequences, enhancing understanding of their combinatorial structure.

Findings

Chebyshev polynomials relate to Fibonacci-like sequences via diagonal sums

A visual column sliding method provides an intuitive proof

Connections extend to Tribonacci, Tetranacci, and higher-order sequences

Abstract

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.