A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS

Abstract

In this paper, we introduce a tiling approach to (p,q)-Fibonacci and (p,q)-Lucas numbers that generalize of the well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal ve Jacobsthal-Lucas numbers. We show that nth (p,q)-Fibonacci number is interpreted as the number of ways to tile a 1×n board with cells labeled 1,2,...,n using colored 1×1 squares and 1×2 dominoes, where there are p kind colors for squares and q kind colors for dominoes. Then nth (p,q)-Lucas number is interpreted as the number of ways to tile a circular 1×n board with squares and dominoes. We also present some generalized Fibonacci and Lucas identities using this tiling approach.