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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## Details

### Authors and affiliations

*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 𝕁_{0}^{(3)} = *a*, 𝕁_{1}^{(3)} = *b* and 𝕁_{2}^{(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

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## Cite this paper

APACerda-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.

ChicagoCerda-Morales, Gamaliel “The Gelin–Cesàro identity in some third-order Jacobsthal sequences.” Notes on Number Theory and Discrete Mathematics 25, no. 2 (2019): 57-67, doi: 10.7546/nntdm.2019.25.2.57-67.

MLACerda-Morales, Gamaliel “The Gelin–Cesàro identity in some third-order Jacobsthal sequences.” Notes on Number Theory and Discrete Mathematics 25.2 (2019): 57-67. Print, doi: 10.7546/nntdm.2019.25.2.57-67.