Generalized Pell Numbers and Some Relations with Fibonacci Numbers

Abstract

In this paper, we generalize the companion Pell sequence. We provide combinatorial, graph, and matrix representations of this sequence.Using these representations, we describe some properties of the generalized Pell numbers and the generalized companion Pell numbers. We define the golden Pell matrix for determining the generalized Pell sequences and, among other results, prove the “generalized Cassini formula” for them.Moreover, we establish some relations between generalized Pell numbers and the classical Fibonacci numbers.