The k-Nacci Sequences in Some Special Modular Groups

Abstract

A k-nacci (k-step Fibonacci) sequence in a finite group is a sequence of group elements x0, x1, x2, …, xn, … for which, given an initial (seed) set x0, x1, x2, …, xj-1, each element is defined by [Formula: see text] From the definition, it is clear that the period of the k-nacci sequence in a group depends on the chosen generating set and the order in which the assignments of x0, x1, x2, …, xj-1 are made. In this paper we examine the periods of the k-nacci sequences in the groups 𝔐2, [Formula: see text] and ℜ2, where each term of the sequence is reduced modulo 2.