On The Usual Fibonacci and Generalized Order-k Pell Numbers

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 …nd families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, F2i 1; and their sums. Also extending these matrix representations, we …nd families of super-diagonal matrices such that the permanents of these matrices are the generalized order-k Pell numbers and their sums. 1. Introduction The well-known Fibonacci sequence fFng is de…ned by the following recursive relation, for n > 2; Fn = Fn 1 + Fn 2: with initial conditions F1 = F2 = 1: The Pell sequence fPng is de…ned recursively by the equation, for n > 2 Pn = 2Pn 1 + Pn 2 (1.1) where P1 = 1; P2 = 2: In [5], Ercolano gave the matrix method for generating the Pell sequence as follows: M = Pn+1 Pn Pn Pn 1 (1.2) The permanent of an n-square matrix A = (aij) is de…ned by