A note on periodic linear recurrence relations

József Bukor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 1–4
DOI: 10.7546/nntdm.2026.32.1.1-4
Full paper (PDF, 173 Kb)

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Authors and affiliations

József Bukor
Department of Informatics, J. Selye University
945 01 Komarno, Slovakia

Abstract

We provide an elementary proof of the fact that a sequence defined by a linear recurrence relation with integer coefficients is periodic if and only if all characteristic roots are distinct roots of unity. Additionally, we discuss the case in which the coefficients of the recurrence relation are restricted to the set {–1,0,1}.

Keywords

• Arbitrary order recurrence relation
• Periodic sequence
• Cyclotomic polynomial

2020 Mathematics Subject Classification

• 11B39

References

1. Atanassov, K. T., & Shannon, A. G. (2025). Two Fibonacci-like sequences. Notes on Number Theory and Discrete Mathematics, 31(2), 335–339.
2. Brookfield, G. (2016). The coefficients of cyclotomic polynomials. Mathematics Magazine, 89(3), 179–188.
3. Everest, G., Van der Poorten, A., Shparlinski, I., & Ward, T. (2003). Recurrence Sequences (Mathematical Surveys and Monographs; Vol. 104). Providence: RI:American Mathematical Society.
4. Greiter, G. (1978). A simple proof for a theorem of Kronecker. The American Mathematical Monthly, 85(9), 756–757.
5. Gryszka, K. (2025). Another six Fibonacci-like sequences. Notes on Number Theory and Discrete Mathematics, 31(3), 563–569.

Manuscript history

• Received: 18 November 2025
• Accepted: 30 January 2026
• Online First: 12 February 2026

Copyright information

Ⓒ 2026 by the Author.
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).