A note on generalized Fibonacci sequences

Consider the generalized Fibonacci sequence {qn}n=0∞ having initial conditions q0=0,q1=1 and recurrence relation qn=aqn-1+qn-2 (when n is even) or qn=bqn-1+qn-2 (when n is odd), where a and b are nonzero real numbers. These sequences arise in a natural way in the study of continued fractions of quadratic irrationals and combinatorics on words or dynamical system theory. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of {qn} with a=b=1. Pell's sequence is {qn} with a=b=2 and the k-Fibonacci sequence is {qn} with a=b=k. In this article, we study numerous new properties of these sequences and investigate a sequence closely related to these sequences which can be regarded as a generalization of Lucas sequence of the first kind.