On Period of Generalized Fibonacci Sequence Over Finite Ring and Tridiagonal Matrix

Abstract

In this study, { } Fibonacci sequence was defined over an arbitrary ring and its some properties are investigated. The terms of this sequence are derivated by Tridiagonal determinant of the matrix. It was shown that this sequence is periodic and their period is obtained. It was shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci sequence in arbitrary rings is periodic. It was seen that order of cyclic group generated with matrix = [ 1 0 ] is equal to the period of this sequence where , are arbitrary elements of the ring. Also, the period of this sequence is compared with Wall number of Fibonacci sequence and it was shown that this period always was an even number.

Applications of Pell Polynomials in Rings

“…Fine gave the classification of all finite rings of order 2 where a prime (Fine 1993). The period of generalized Fibonacci sequence in arbitrary rings has been obtained by Taşyurdu and Dilmen (Taşyurdu & Dilmen, 2017). Also, they showed that determinant of Tridiagonal matrix derivated the terms of this sequence.…”

“…Also, they showed that determinant of Tridiagonal matrix derivated the terms of this sequence. Taşyurdu and Deveci study the Fibonacci polynomials in the ring of complex numbers and modulo m (Taşyurdu & Deveci 2017). If a sequence consists only of a fixed subsequence after a certain point, this sequence is called a periodic sequence.…”