H. Doostie and C. M. Campell
 

Abstract
The Fibonacci length of a finitely generated finite group $G = \langle a, b \rangle$ is the least integer $n$ such that, for the sequences $x_1 = a$, $x_2 = b$, $x_{i+2} = x_i x_{i +1}$, $(i > 1)$ of the elements of $G$, $x_{n+1} = x_1$ and $x_{n+2} = x_2$. The group $D_{2n}$, $Q_{2n}$ and the simple groups of order $\le 10^5$ are the only known groups that their Fibonacci lengths have been known. In this paper we shall generalize this notion for the 3-generated groups and whereby we calculate the Fibonacci lengths of the groups Aut$(D_{2n})$ and $Aut$(Q_{2^n})$ which involve certain sequences of Tribonacci numbers.