On the Third-Order Jacobsthal and Third-Order Jacobsthal–Lucas Sequences and Their Matrix Representations

  • Gamaliel Cerda-MoralesEmail author


In this paper, we first give new generalizations for third-order Jacobsthal \(\{J_{n}^{(3)}\}_{n\in \mathbb {N}}\) and third-order Jacobsthal–Lucas \(\{j_{n}^{(3)}\}_{n\in \mathbb {N}}\) sequences for Jacobsthal and Jacobsthal–Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of \(\{J_{n}^{(3)}\}_{n\in \mathbb {N}}\) and \(\{j_{n}^{(3)}\}_{n\in \mathbb {N}}\). Then, we investigate their properties.


Generalized Fibonacci number third-order Jacobsthal number third-order Jacobsthal–Lucas number matrix representation matrix methods generalized Jacobsthal number 

Mathematics Subject Classification

11B37 11B39 15A15 



The author also thanks the suggestions sent by the reviewer, which have improved the final version of this article.


  1. 1.
    Barry, P.: Triangle geometry and Jacobsthal numbers. Irish. Math. Soc. Bull. 51, 45–57 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cerda-Morales, G.: Identities for third order Jacobsthal quaternions. Adv. Appl. Clifford Algebra 27(2), 1043–1053 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cerda-Morales, G.: On a generalization of Tribonacci quaternions. Mediter. J. Math. 14(239), 1–12 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Civciv, H., Türkmen, R.: On the \((s, t)\)-Fibonacci and Fibonacci matrix sequences. ARS Combin. 87, 161–173 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Civciv, H., Türkmen, R.: Notes on the \((s, t)\)-Lucas and Lucas matrix sequences. ARS Combin. 89, 271–285 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cook, C.K., Bacon, M.R.: Some identities for Jacobsthal and Jacobsthal–Lucas numbers satisfying higher order recurrence relations. Ann. Math. Inf. 41, 27–39 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Horadam, A.F.: Jacobsthal representation numbers. Fibonacci Quart. 34, 40–54 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticasPontificia Universidad Católica de ValparaísoValparaisoChile

Personalised recommendations