On r-Jacobsthal and r-Jacobsthal-Lucas numbers


Recently, Bród introduced a new Jacobsthal-type sequence which is called r-Jacobsthal sequence in current study. After defining the appropriate r-Jacobsthal–Lucas sequence for the r-Jacobsthal sequence, we obtain some properties of these two sequences. For simpler results, we define two new sequences and examine their properties, too. Finally, we generalize some well-known identities.


r-Jacobsthal numbers; r-Jacobsthal–Lucas numbers; Binet formula

D. Bród, On a new Jacobsthal-type sequence, Ars Combin. 150 (2020), 21–29.

A. Daşdemir, The representation, generalized Binet formula and sums of the generalized Jacobsthal p-sequence, Hittite J. Sci. Eng. 3 (2016), no. 2, 99–104.

L.E. Dickson, History of the Theory of Numbers. Vol. I: Divisibility and Primality, Chelsea Publishing Co., New York, 1952.

M. Edson and O. Yayenie, A new generalization of Fibonacci sequence & extended Binet’s formula, Integers 9 (2009), no. 6, 639–654.

S. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Statistics 2 (2014), no. 1, 67–77.

A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), no. 3, 161–176.

A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), no. 1, 40–54.

D. Jhala, K. Sisodiya, and G.P.S. Rathore, On some identities for k-Jacobsthal numbers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 12, 551–556.

R.E. Merrifield and H.E. Simmons, Topological Methods in Chemistry, John Wiley & Sons, New York, 1989.

S. Uygun, The (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequences, Appl. Math. Sci. (Ruse) 9 (2015), no. 70, 3467–3476.

S. Uygun and E. Owusu, A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences), J. Math. Anal. 7 (2016), no. 5, 28–39.


Published : 2023-02-07

BilgiciG., & BródD. (2023). On r-Jacobsthal and r-Jacobsthal-Lucas numbers. Annales Mathematicae Silesianae, 37(1), 16-31. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/15208

Göksal Bilgici  gbilgici@kastamonu.edu.tr
Elementary Mathematics Education, Kastamonu University  Turkey
Dorota Bród 
Zakład Matematyki Dyskretnej, Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska  Poland

Creative Commons License

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

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.

  1. 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.
  2. 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.
  3. 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.
  4. 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.