Relations between Generalized Bi-Periodic Fibonacci and Lucas Sequences

Younseok Choo

doi:10.3390/math8091527

Abstract

In this paper we consider a generalized bi-periodic Fibonacci {fn} and a generalized bi-periodic Lucas sequence {qn} which are respectively defined by f0=0, f1=1, fn=afn−1+cfn−2 (n is even) or fn=bfn−1+cfn−2 (n is odd), and q0=2d, q1=ad, qn=bqn−1+cqn−2 (n is even) or qn=afn−1+cqn−2 (n is odd). We obtain various relations between these two sequences.

Highlights

As is well known, the Fibonacci sequence { Fn } is generated from the recurrence relationFn = Fn−1 + Fn−2 with initial condition F0 = 0, F1 = 1, and the Lucas sequence { Ln } satisfies the same recurrence relation with initial condition L0 = 2, L1 = 1
If n is odd Recently Tan and Leung [18] studied the properties of the bi-periodic Fibonacci and Lucas sequences, and stated that the following identities can be obtained using matrix method in [19]: f m gn + f n gm = 2
In this paper we considered two kinds of general sequences, i.e., generalized bi-periodic Fibonacci sequence and generalized bi-periodic Lucas sequence

Summary

Introduction

The Fibonacci sequence { Fn } is generated from the recurrence relation Many authors generalized the Fibonacci and Lucas sequences by changing initial conditions and/or recurrence relations. Introduced a bi-periodic Fibonacci sequence { pn } defined by p0 = 0, p1 = 1, pn = In this paper we consider a generalized bi-periodic Fibonacci sequence { f n } and a generalized bi-periodic Lucas sequence {qn } which are generalizations of { pn } and {ln } respectively, and are defined as

Results

Conclusion