A Generalized Binet Formula That Counts the Tilings of a $(2\times n)$-board

Abstract

In this paper, we answer an open problem raised by Katz and Stenson. They considered the tilings of a $(2\times n)$-board using $a$ colors of squares and $b$ colors of dominoes. The number of such tilings, denoted by $K_n^{a,b}$, is a generalization of the Fibonacci numbers. Obtaining a Binet-style formula for these numbers is such problem. We obtain a generalized Binet formula for $K_^{a,b}$.