The Generalized Order-k Lucas Sequences in Finite Groups

We study the generalized order-k Lucas sequences modulo m. Also, we define the ith generalized order-k Lucas orbit l(A)(i,{alpha 1,alpha 2, ... ,alpha k-1}) (G) with respect to the generating set A and the constants alpha(1), alpha(2), and alpha(k-1) for a finite group G = < A >. Then, we obtain the lengths of the periods of the ith generalized order-k Lucas orbits of the binary polyhedral groups < n, 2, 2 >, < 2, n, 2 >, < 2, 2, n > and the polyhedral groups (n, 2, 2), (2, n, 2), (2, 2, n) for 1 <= i <= k.