The generalized order- k Fibonacci–Pell sequence by matrix methods

Abstract

In this paper, we consider the usual and generalized order- k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order- k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order- k F–P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order- k Pell numbers, as well as the Pell numbers themselves.