On The Usual Fibonacci and Generalized Order-\(k\) Pell Numbers

Abstract

In this paper, we give some relations involving the usual Fibonacci and generalized order-\(k\) Pell numbers. These relations show that the generalized order-\(k\) Pell numbers can be expressed as the summation of the usual Fibonacci numbers. We find families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, \(F_{2i-1}\), and their sums. Also, extending these matrix representations, we find families of super-diagonal matrices such that the permanents of these matrices are the generalized order-\(k\) Pell numbers and their sums.