On the k-Nacci Sequences in Finite Binary Polyhedral Groups.

A k-nacci sequence in a finite group is a sequence of group elements x0,x1,...,xn,... for which, given an initial (seed) set x0,x1,...,xj-1, each element is defined by \[ x_{n}= \left\{\begin{array}{@{}l@{\quad}l@{}} x_{0}x_{1}\cdots x_{n-1}&\mbox{ for }j \le n<k,\\ x_{n-k}x_{n-k+1}\cdots x_{n-1}&\mbox{ for } n\ge k. \end{array}\right. \] It is important to note that the Fibonacci length depends on the chosen generating n-tuple for a group. The binary polyhedral groups have been studied recently by C.M. Campbell and P.P. Campbell for their Fibonacci lengths. In this paper, we obtain the period of k-nacci sequences in the binary polyhedral groups 〈2,2,2〉, 〈n,2,2〉, 〈2,n,2〉 and 〈2,2,n〉 for any n > 2. [ABSTRACT FROM AUTHOR]