A Study of Generalized Fourth-Order Pell Sequences

Sequences related to the Pell generalized equation

We consider sequences of the type An = 6An−1 − An−2, A0 = r, A1 = s (r and s integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type. Let α = 3 + 2 √ 2, β = 3 − 2 √ 2 be the roots of the polynomial x − 6x+ 1. Note that α+β = 6, αβ = 1, α−β = 4 √ 2. Also let γ = 1+ √ 2...

Some Properties of Generalized Third Order Pell Numbers

This paper considers some properties of the third order recursive sequence defined by the linear recurrence relation , 2 , 1 , 0 , 3 , 2 3 , 2 , , = ≥ + = − − m n w w w n m n m m n m , with appropriate initial conditions. The present work follows on from the case m = 0 (Shannon et al). Relationships with the well-known sequences of Fibonacci, Lucas and Pell are developed. The motivation for the...

The k-nacci Sequences and The Generalized Order-k Pell Sequences in The Semi-Direct Product of Finite Cyclic Groups

On The Generalized Fibonacci And Pell Sequences By Hessenberg Matrices

In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de…ned by the recurrence...

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 mat...