Generalized Bronze Leonardo sequence

Engin Özkan and Hakan Akkuş
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 811–824
DOI: 10.7546/nntdm.2024.30.4.811-824
Full paper (PDF, 624 Kb)

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Authors and affiliations

Engin Özkan
Department of Mathematics, Faculty of Science, Marmara University, Istanbul, Türkiye

Hakan Akkuş
Department of Mathematics, Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan, Türkiye

Abstract

In this study, we define the Bronze Leonardo, Bronze Leonardo–Lucas, and Modified Bronze Leonardo sequences, and some terms of these sequences are given. Then, we give special summation formulas, special generating functions, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence’s generating functions, and the third is with the help of the matrices. In addition, we find the special relations between the terms of the Bronze Leonardo, Bronze Leonardo–Lucas, and Modified Bronze Leonardo sequences. Moreover, we examine the relationships among the Bronze Fibonacci and Bronze Lucas sequences of these sequences. Finally, we associate these sequences with the matrices.

Keywords

• Leonardo number
• Generating function
• Binet formula
• Fibonacci sequence

2020 Mathematics Subject Classification

• 11B39
• 11B83
• 05A15

References

1. Akbiyik, M., & Alo, J. (2021). On third-order Bronze Fibonacci numbers. Mathematics, 9(20), Article ID 2606.
2. Alo, J. (2022). On third order Bronze Fibonacci quaternions. Turkish Journal of Mathematics and Computer Science, 14(2), 331–339.
3. Alp, Y., & Koçer, E. G. (2021). Some properties of Leonardo numbers. Konuralp Journal of Mathematics, 9(1), 183–189.
4. Avazzadeh, Z., Hassani, H., Agarwal, P., Mehrabi, S., Javad Ebadi, M., & Hosseini Asl, M. K. (2023). Optimal study on fractional fascioliasis disease model based on generalized Fibonacci polynomials. Mathematical Methods in the Applied Sciences, 46(8), 9332–9350.
5. Aydınyüz, S., & Aşcı, M. (2023). The Moore–Penrose Inverse of the Rectangular Fibonacci Matrix and Applications to the Cryptology. Advances and Applications in Discrete Mathematics, 40(2), 195–211.
6. Catarino, P. M., & Borges, A. (2019). On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75–86.
7. Catarino, P., & Ricardo, S. (2022). A note on special matrices involving k-Bronze Fibonacci numbers. In: International Conference on Mathematics and Its Applications in Science and Engineering, 135–145.
8. Catarino, P., & Ricardo, S. (2022). On quaternion Gaussian Bronze Fibonacci numbers. Annales Mathematicae Silesianae, 36(2), 129–150.
9. Çelik, S., Durukan, İ., & Özkan, E. (2021). New recurrences on Pell numbers, Pell–Lucas numbers, Jacobsthal numbers, and Jacobsthal–Lucas numbers. Chaos, Solitons & Fractals, 150, Article ID 111173.
10. De Oliveira, R. R., & Alves, F. R. V. (2019). An investigation of the bivariate complex Fibonacci polynomials supported in didactic engineering: an application of Theory of Didactics Situations (TSD). Acta Scientiae, 21(3), 170–195.
11. Griggs, J. R., Hanlon, P., Odlyzko, A. M., & Waterman, M. S. (1990). On the number of alignments of k sequences. Graphs and Combinatorics, 6(2), 133–146.
12. Karaaslan, N. (2022). Gaussian bronze Lucas numbers. Bilecik Şeyh Edebali Üniversitesi Fen Bilimleri Dergisi, 9(1), 357–363.
13. Kiliç, E., & Stanica, P. (2011). A matrix approach for general higher order linear recurrences. Bulletin of the Malaysian Mathematical Sciences Society, 34(1), 51–67.
14. Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications, 2. John Wiley & Sons, New Jersey.
15. Kuloǧlu, B., Eser, E., & Özkan, E. (2023). On the properties of r-circulant matrices involving Generalized Fermat numbers. Sakarya University Journal of Science, 27(5), 956–965.
16. Otto, H. H. (2022). Fibonacci stoichiometry and superb performance of Nb16W5O55 and related super-battery materials. Journal of Applied Mathematics and Physics, 10(6), 1936–1950.
17. Soykan, Y. (2021). Generalized Oresme Numbers. Earthline Journal of Mathematical Sciences, 7(2), 333–367.
18. Turner, H. A., Humpage, M., Kerp, H., & Hetherington, A. J. (2023). Leaves and sporangia developed in rare non-Fibonacci spirals in early leafy plants. Science, 380(6650), 1188–1192.

Manuscript history

• Received: 17 June 2024
• Revised: 27 November 2024
• Accepted: 28 November 2024
• Online First: 28 November 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).