The Tilings of a $(2\times n)$-Board and Some New Combinatorial Identities

Abstract

We know that the Fibonacci numbers count the tilings of a (1 × n)-board by squares and dominoes, or equivalently, the number of tilings of a (2 × n)-board by dominoes. We use the tilings of a (2 × n)-board by colored unit squares and dominoes to obtain some new combinatorial identities. They are generalization of some known combinatorial identities and in the special case give us the Fibonacci identities.