**Timmy Ma, Richard Vernon and Gurdial Arora**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 1, Pages 67–80

DOI: 10.7546/nntdm.2024.30.1.67-80

**Full paper (PDF, 256 Kb)**

## Details

### Authors and affiliations

Timmy Ma

*Department of Mathematics, Xavier University of Louisiana
1 Drexel Drive, New Orleans, LA 70125, United States*

Richard Vernon

*Department of Mathematics, Xavier University of Louisiana
1 Drexel Drive, New Orleans, LA 70125, United States*

Gurdial Arora

*Department of Mathematics, Xavier University of Louisiana
1 Drexel Drive, New Orleans, LA 70125, United States*

### Abstract

We will explore the generalization of the four different 2-Fibonacci sequences defined by Atanassov. In particular, we will define recurrence relations to generate each part of a 2-Fibonacci sequence, discuss the generating function and Binet formula of each of these sequences, and provide the necessary and sufficient conditions to obtain each type of Binet formula.

### Keywords

- Fibonacci sequences
- Binet formula
- Bi-periodic Fibonacci sequence
- 2-Fibonacci sequences

### 2020 Mathematics Subject Classification

- 11B39

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### Manuscript history

- Received: 24 September 2023
- Revised: 26 February 2024
- Accepted: 27 February 2024
- Online First: 28 February 2024

### Copyright information

Ⓒ 2024 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

- Atanassov, K. (2006). A new direction of Fibonacci sequence modification.
*Notes on Number Theory and Discrete Mathematics*, 12(1), 25–32. - Atanassov, K. (2010). Combined 2-Fibonacci sequences.
*Notes on Number Theory and Discrete Mathematics*, 16(2), 24–28. - Atanassov, K. (2014). A set of Lucas sequences.
*Notes on Number Theory and Discrete Mathematics*, 20(2), 1–5. - Atanassov, K. (2022). Two 2-Fibonacci sequences generated by a mixed scheme. Part 1.
*Notes on Number Theory and Discrete Mathematics*, 28(2), 331–338.

## Cite this paper

Ma, T., Vernon. R., & Arora, G. (2024). Generalization of the 2-Fibonacci sequences and their Binet formula. *Notes on Number Theory and Discrete Mathematics*, 30(1), 67-80, DOI: 10.7546/nntdm.2024.30.1.67-80.