The GCD sequences of the altered Lucas sequences
Abstract
In this study, we give two sequences {Ln+}n≥1 and {Ln-}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 Ln+ and Ln- 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.
Keywords
Fibonacci and Lucas numbers; altered Lucas sequence; r-consecutive GCD sequence
References
2. U. Dudley and B. Tucker, Greatest common divisors in altered Fibonacci sequences, Fibonacci Quart. 9 (1971), no. 1, 89–91.
3. L. Hajdu and M. Szikszai, On the GCD-s of k consecutive terms of Lucas sequences, J. Number Theory 132 (2012), no. 12, 3056–3069.
4. S. Hernández and F. Luca, Common factors of shifted Fibonacci numbers, Period. Math. Hungar. 47 (2003), no. 1-2, 95–110.
5. L. Jones, Primefree shifted Lucas sequences, Acta Arith. 170 (2015), no. 3, 287–298.
6. T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
7. A. Rahn and M. Kreh, Greatest common divisors of shifted Fibonacci sequences revisited, J. Integer. Seq. 21 (2018), no. 6, Art. 18.6.7, 12 pp.
8. J. Spilker, The GCD of the shifted Fibonacci sequence, in: J. Sander et al. (eds.), From Arithmetic to Zeta-Functions, Springer, Cham, 2016, pp. 473–483.
Eregli Kemal Akman Vocational School, Necmettin Erbakan University, Turkey Turkey
https://orcid.org/0000-0002-8304-9525
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
