The Gelin–Cesàro identity in some third-order Jacobsthal sequences

Gamaliel Cerda-Morales
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 57-67
DOI: 10.7546/nntdm.2019.25.2.57-67
Full paper (PDF, 173 Kb)

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Authors and affiliations

Gamaliel Cerda-Morales

Departamento de Matematica, Universidad de Concepcion Esteban Iturra s/n,
Barrio Universitario, Concepcion, Chile

Abstract

In this paper, we deal with two families of third-order Jacobsthal sequences. The first family consists of generalizations of the Jacobsthal sequence. We show that the Gelin–Cesàro identity is satisfied. Also, we define a family of generalized third-order Jacobsthal sequences {𝕁_n⁽³⁾}_{n ≥ 0} by the recurrence relation

𝕁_n+3⁽³⁾ = 𝕁_n+2⁽³⁾ + 𝕁_n+1⁽³⁾ + 2𝕁_n⁽³⁾, n ≥ 0,

with initials conditions 𝕁₀⁽³⁾ = a, 𝕁₁⁽³⁾ = b and 𝕁₂⁽³⁾ = c, where a, b and c are non-zero real numbers. Many sequences in the literature are special cases of this sequence. We find the generating function and Binet’s formula of the sequence. Then we show that the Cassini and Gelin–Cesàro identities are satisfied by the indices of this generalized sequence.

Keywords

• Third-order Jacobsthal sequence
• Generating function
• Jacobsthal sequence
• Generalized third-order Jacobsthal sequence

2010 Mathematics Subject Classification

• 11B39
• 05A15

References

1. Barry, P. (2003). Triangle geometry and Jacobsthal numbers, Irish Math. Soc. Bull., 51, 45–57.
2. Cerda-Morales, G. (2017). Identities for Third Order Jacobsthal Quaternions, Advances in Applied Clifford Algebras, 27 (2), 1043–1053.
3. Cerda-Morales, G. (2017). On a Generalization of Tribonacci Quaternions, Mediterranean Journal of Mathematics, 14:239, 1–12.
4. Cerda-Morales, G. (2018). Dual Third-order Jacobsthal Quaternions, Proyecciones Journal of Mathematics, 37 (4), 731-747.
5. Cook, C. K., & Bacon, M. R. (2013). Some identities for Jacobsthal and Jacobsthal–Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41, 27–39.
6. Dickson, L. E. (1966). History of the Theory of Numbers, Vol. I, Chelsea Publishing Co., New York.
7. Horadam, A. F. (1988). Jacobsthal and Pell Curves, The Fibonacci Quarterly, 26 (1), 79–83.
8. Horadam, A. F. (1996). Jacobsthal representation numbers, The Fibonacci Quarterly, 43 (1), 40–54.
9. Melham, R. S., & Shannon, A. G. (1995). A generalization of the Catalan identity and some consequences, The Fibonacci Quarterly, 33, 82–84.
10. Sahin, M. (2011). The Gelin–Cesàro identity in some conditional sequences, Hacettepe Journal of Mathematics and Statistics, 40 (6), 855–861.