The Hadamard-type k-step Fibonacci sequences in groups

Abstract

In this paper, we give the relationships between the orders of the cyclic groups obtained from the generating matrices of the Hadamard-type k-step Fibonacci sequences and the periods and the ranks of the Hadamard-type k-step Fibonacci sequences modulo m. We also extend the Hadamard-type k-step Fibonacci sequences to groups and then we investigate the structure of groups which have two or three generators by the aid of these sequences. Furthermore, we determine the periods and the ranks of the Hadamard-type k-step Fibonacci sequences in finite groups using the basic Hadamard-type k-step Fibonacci sequences. Finally, we obtain the periods, the basic periods and the ranks of the Hadamard-type 3-step Fibonacci sequences in the dihedral group (Formula presented.) and the generalized quaternion group (Formula presented.) with respect to the generating pairs (a, b) and (Formula presented.).