K-nacci Sequences in Finite Triangle Groups

A k-nacci sequence in a finite group is a sequence of group elements x(0), x(1), x(2), ... , x(n), ... for which, given an initial (seed) set x(0), x(1), x(2), ... , x(j-1), each element is defined by x(n) = x(0)x(1) ... x(n-1), for j <= n < k, and x(n) = x(n-k)x(n-k+1) ... x(n-1), for n >= k. We also require that the initial elements of the sequence, x(0), x(1), x(2), ... , x(j-1), generate the group, thus forcing the k-nacci sequence to reflect the structure of the group. The K-nacci sequence of a group generated by x(0), x(1), x(2), ... , x(j-1) is denoted by F-k(G; x(0), x(1), ..., x(j-1)) and its period is denoted by P-k(G; x(0), x(1), ... , x(j-1). In this paper, we obtain the period of K-nacci sequences in finite polyhedral groups and the extended triangle groups. Copyright (C) 2009 E. Karaduman and O. Deveci.