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
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Authors and affiliations

Gamaliel Cerda-Morales

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


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 {𝕁n(3)}n ≥ 0 by the recurrence relation

𝕁n+3(3) = 𝕁n+2(3) + 𝕁n+1(3) + 2𝕁n(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.


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

2010 Mathematics Subject Classification

  • 11B39
  • 05A15


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Cite this paper

Cerda-Morales, Gamaliel (2019). The Gelin–Cesàro identity in some third-order Jacobsthal sequences. Notes on Number Theory and Discrete Mathematics, 25(2), 57-67, doi: 10.7546/nntdm.2019.25.2.57-67.

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