The authors consider finitely generated groups. If is a finite generating system of the group , then the Fibonacci orbit of with respect to , denoted by , and the Fibonacci length of with respect to , denoted by or , are defined in the usual manner.
In this paper, the authors examine the Fibonacci length of certain classes of 2-generator metacyclic groups including the metacyclic Fox groups , . They also study the Fibonacci length of the groups when is odd. They prove that and find when . In case is even the length is known by a result due to D. D. Wall (1960).