<abstract xmlns="http://www.w3.org/1999/xhtml">In this study, we give two sequences <italic>{L</italic>+<italic>n}n≥</italic>1 and <italic>{L−n}n≥</italic>1 derived by altering the Lucas numbers with {<italic>±</italic>1, <italic>±</italic>3}, terms of which are called as altered Lucas numbers. We give relations connected with the Fibonacci <italic>Fn</italic> and Lucas <italic>Ln</italic> numbers, and construct recurrence relations and Binet’s like formulas of the <italic>L</italic>+<italic>n</italic> and <italic>L−n</italic> 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 (<italic>GCD</italic>) of <italic>r</italic>-consecutive altered Lucas numbers. We obtain <italic>r</italic>-consecutive <italic>GCD</italic> sequences according to the altered Lucas numbers, and show that their <italic>GCD</italic> sequences are unbounded or periodic in terms of values <italic>r</italic>.</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.

The GCD Sequences of the Altered Lucas Sequences

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"The GCD Sequences of the Altered Lucas Sequences"}

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...

The GCD Sequences of the Altered Lucas Sequences

Published Online: Jun 22, 2020

Page range: 222 - 240

Received: Jun 19, 2019

Accepted: Mar 16, 2020

DOI: https://doi.org/10.2478/amsil-2020-0005

Keywords
Fibonacci and Lucas numbers, altered Lucas sequence, -consecutive sequence

© 2020 Fikri Koken, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

The GCD Sequences of the Altered Lucas Sequences

Published Online: Jun 22, 2020

Page range: 222 - 240

Received: Jun 19, 2019

Accepted: Mar 16, 2020

DOI: https://doi.org/10.2478/amsil-2020-0005

KeywordsFibonacci and Lucas numbers, altered Lucas sequence, -consecutive sequence

© 2020 Fikri Koken, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Fibonacci and Lucas numbers, altered Lucas sequence, -consecutive sequence