The GCD Sequences of the Altered Lucas Sequences

Fikri Koken

doi:10.2478/amsil-2020-0005

Abstract

Abstract In this study, we give two sequences {L + n}n≥ 1 and {L− n}n≥ 1 derived by altering the Lucas numbers with {±1, ±3}, terms of which are called as altered Lucas numbers. We give relations connected with the Fibonacci Fn and Lucas Ln numbers, and construct recurrence relations and Binet’s like formulas of the L + n and L− n numbers. It is seen that the altered Lucas numbers have two distinct factors from the Fibonacci and Lucas sequences. Thus, we work out the greatest common divisor (GCD) of r-consecutive altered Lucas numbers. We obtain r-consecutive GCD sequences according to the altered Lucas numbers, and show that their GCD sequences are unbounded or periodic in terms of values r.

Highlights

Let Fn and Ln denote nth Fibonacci and Lucas numbers, respectively
We show that the altered Lucas numbers L+n and L−n satisfy interrelationships with the Fibonacci and Lucas numbers
As an alternative method to the definitions given in (2.1), (2.2) and all results of Theorem 2.1, we investigate a Binet’s like formula, which is commonly used in the proof of the properties of the integer sequences

Summary

Introduction

Let Fn and Ln denote nth Fibonacci and Lucas numbers, respectively. The numbers Fn and Ln, are entries of sequences {Fn}n≥0 and {Ln}n≥0, are given by the linear recurrence relations,. In [5], the author studies two shifted sequences Ua ± k of the Lucas sequences of the first kind, where Ua = {un}n≥0, a ∈ Z, un = aun−1 + un−2 for n ≥ 2, u0 = 0, u1 = 1, and shows that there exist infinitely many integers k such that two sequences are prime free. This result extends previous works for the shifted Fibonacci sequences, when a = 1 and k = 1. We establish some r-consecutive GCD shifted sequences from two altered Lucas sequences, and give some properties of them

The altered Lucas sequences

Properties of the GCD sequences of the altered Lucas sequences

Conclusion