Generalization of the 2-Fibonacci sequences and their Binet formula

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)

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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

References

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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).