A study of the k-step generalized balancing sequences

Abstract

In this paper, firstly, we define the k-step generalized Balancing sequences and study the Binet formula of these sequences. Also, we find families of super-diagonal matrices such that the permanents of these matrices are the elements of the k-step generalized Balancing sequences. Finally, we examine the periods of the k-step Balancing sequences in the semi-direct product presented by G = < x, y | x2m−1 = y2 = 1, yxy = x−1 >  for the generating pair (x, y).