The generalized Horadam model and the fundamental sequence of a nonhomogeneous linear recursive relation

Mustapha Rachidi, Irene Magalhães Craveiro and Elen Viviani Pereira Spreafico
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 173–186
DOI: 10.7546/nntdm.2026.32.1.173-186
Full paper (PDF, 301 Kb)

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Authors and affiliations

Mustapha Rachidi
INMA, Federal University of Mato Grosso do Sul – UFMS
Campo Grande, MS, Brazil

Irene Magalhães Craveiro
FACET – Federal University of Grande Dourados – UFGD
Dourados, MS, Brazil

Elen Viviani Pereira Spreafico
INMA, Federal University of Mato Grosso do Sul – UFMS
Campo Grande, MS, Brazil

Abstract

This study concerns some properties of the generalized nonhomogeneous Horadam model and its application to four types of known sequences of numbers. Regarding the four sequences of this model, we provide results on combinatorial and linear expressions, and also on the analytical representation of Binet. Our approach is based on results we have developed regarding the general setting of nonhomogeneous linear recursive sequences, especially the fundamental sequence related to their homogeneous part.

Keywords

• Generalized nonhomogeneous Horadam model
• Nonhomogeneous linear recurrence relation
• Fundamental Fibonacci system
• Fundamental sequence

2020 Mathematics Subject Classification

• 15A99
• 40A05
• 40A25
• 45M05

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Manuscript history

• Received: 7 October 2025
• Revised: 3 March 2026
• Accepted: 9 March 2026
• Online First: 10 March 2026

Copyright information

Ⓒ 2026 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).