Relations between Generalized Bi-Periodic Fibonacci and Lucas Sequences

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.

Keywords

• generalized bi-periodic Fibonacci sequence
• generalized bi-periodic Lucas sequence
• Binet’s formula