## Abstract

Consider the Fibonacci sequence having initial conditions *F*
_{0} = 0, *F*
_{1} = 1 and recurrence relation *F _{n}
* =

*F*

_{n–1}+

*F*

_{n–2}(

*n*≥ 2). The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this article, we study a new generalization {

*q*}, with initial conditions

_{n}*q*

_{0}= 0 and

*q*

_{1}= 1 which is generated by the recurrence relation

*q*=

_{n}*aq*

_{n–1}+

*q*

_{n–2}(when

*n*is even) or

*q*=

_{n}*bq*

_{n–1}+

*q*

_{n–2}(when

*n*is odd), where

*a*and

*b*are nonzero real numbers. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of {

*q*} with

_{n}*a*=

*b*= 1. Pell's sequence is {

*q*} with

_{n}*a*=

*b*= 2 and the

*k*-Fibonacci sequence is {

*q*} with

_{n}*a*=

*b*=

*k*. We produce an extended Binet's formula for the sequence {

*q*} and, thereby, identities such as Cassini's, Catalan's, d'Ocagne's, etc.

_{n}
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