On a generalization of the Hosoya triangle

Turhan Çifçi, Hamza Menken and Orhan Dişkaya
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 187–197
DOI: 10.7546/nntdm.2026.32.1.187-197
Full paper (PDF, 342 Kb)

Details

Authors and affiliations

Turhan Çifçi
Department of Mathematics, Mersin University
Mersin, Türkiye

Hamza Menken
Department of Mathematics, Mersin University
Mersin, Türkiye

Orhan Dişkaya
Department of Mathematics, Mersin University
Mersin, Türkiye

Abstract

This paper introduces the Fibonacci polynomial triangle, inspired by the structure of the Hosoya triangle and constructed using Fibonacci polynomials. We then present and rigorously prove a series of novel identities and fundamental properties specifically associated with this Fibonacci polynomial triangle. These findings contribute to a deeper understanding of the algebraic structures and combinatorial patterns that emerge when Fibonacci polynomials are organized in such a triangular fashion, revealing new relationships and characteristics within this framework. This exploration aims to further elucidate the rich interplay between polynomial sequences and triangular constructions.

Keywords

• Fibonacci polynomials
• Lucas polynomials
• Hosoya triangle

2020 Mathematics Subject Classification

• 11B39
• 05A10
• 05A19
• 11B83

References

1. Benjamin, A. T., & Elizondo, D. (2022). Counting on Hosoya’s triangle. The Fibonacci Quarterly, 60(5), 47-55.
2. Blair, M., Flórez, R., & Mukherjee, A. (2019). Matrices in the Hosoya triangle.  The Fibonacci Quarterly, 57(5), 15-28.
3. Czabarka, É., Flórez, R., & Junes, L. (2015). A discrete convolution on the generalized Hosoya triangle. Journal of Integer Sequences, 18(1), Article ID 15.1.6.
4. Dişkaya, O., & Menken, H. (2020). On the Padovan triangle. Journal of Contemporary Applied Mathematics, 10(2), 77–83.
5. Flórez, R., Higuita, R. A., & Junes, L. (2014). GCD property of the generalized Star of David in the generalized Hosoya triangle. Journal of Integer Sequences, 17(3), Article ID 14.3.6.
6. Flórez, R., Higuita, R. A., & Mukherjee, A. (2014). Alternating sums in the Hosoya polynomial triangle. Journal of Integer Sequences, 17(9), Article ID 14.9.5.
7. Flórez, R., Higuita, R. A., & Mukherjee, A. (2018). The Star of David and other patterns in Hosoya polynomial triangles. Journal of Integer Sequences, 21(4), Article ID 18.4.6.
8. Flórez, R., & Junes, L. (2012). GCD properties in Hosoya’s triangle. The Fibonacci Quarterly, 50(2), 163–174.
9. Hoggatt Jr, V. E. (1970). Convolution triangles for generalized Fibonacci numbers. The Fibonacci Quarterly, 8(2), 158–171.
10. Hosoya, H. (1976). Fibonacci triangle. The Fibonacci Quarterly, 14(2), 173–179.
11. Klavžar, S., & Peterin, I. (2007). Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle. Publicationes Mathematicae Debrecen, 71(3–4), 267–278.
12. Koshy, T. (2018). Fibonacci and Lucas Numbers with Applications. Volume 1. John Wiley & Sons, New Jersey.
13. Koshy. T. (2019). Fibonacci and Lucas numbers with applications. Volume 2. John Wiley & Sons, New Jersey.
14. Kuloğlu, B., & Özkan, E. (2022). New Narayana triangle. Journal of Science and Arts, 22(3), 563–570.
15. Šána, J. (1983). Lucas triangle. The Fibonacci Quarterly, 21(3), 192–195.
16. Vincenzi, G., & Siani, S. (2014). Fibonacci-like sequences and generalized Pascal’s triangles. International Journal of Mathematical Education in Science and Technology, 45(4), 609–614.
17. Wells, D. L. (1996). Residue counts modulo three for the Fibonacci triangle. In: Applications of Fibonacci Numbers, 521–536. Springer, Dordrecht.
18. Wilson, B. (1998). The Fibonacci triangle modulo p. The Fibonacci Quarterly, 36, 194–203.

Manuscript history

• Received: 10 June 2025
• Revised: 5 January 2026
• Accepted: 12 March 2026
• Online First: 18 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).