A study on the k-step generalized balancing sequences

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 pre-sented by (formula presented) for the generating pair (x, y).