A k-nacci sequence in a finite group is a sequence of group elements x0 , x1, x2 ,, xn , forwhich, given an initial (seed) set 0 1 2 1 , , , ,j x x x x , each element is defined by0 1 11 1for ,for .nnn k n k nxx x j n kxx x x n k In this paper, we examine the periods of the k-nacci sequences in Miller’s generalization of the polyhedralgroups 2,2 2;q , n,2 2;q , 2, n 2;q , 2,2 n;q , for any n 2 .