On companion sequences associated with Leonardo quaternions : Applications over finite fields
Diana Savindiana.savin@unitbv.roFaculty of Mathematics and Computer ScienceTransilvania University of Brasov500091, RomaniaView full profile → , *Elif TanCorresponding authoretan@ankara.edu.trDepartment of MathematicsFaculty of SciencesAnkara UniversityTandogan, Ankara, 06100, TürkiyeView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 10 Apr 2024
- Published Online:
- 06 Feb 2025
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIOS-1836
- Pages:
- 1569–1585
Abstract
Keywords
Subject Classifications
References
[1] S. Abbad, H. Belbachir, and B. Benzaghou, “Companion sequences associated to the r-Fibonacci sequence: algebraic and combinatorial properties,” Turkish J. Math., vol. 43, no. 3, pp. 1095–1114 (2019).
[2] Y. Alp and E. G. Kocer, “Hybrid Leonardo numbers,” Chaos Solitons Fractals, vol. 150, p. 111128 (2021).
[3] P. Catarino and A. Borges, “On Leonardo numbers,” Acta Math. Univ. Comenianae, vol. 89, no. 1, pp. 75–86 (2019).
[4] A. Dagdeviren, F. Kürüz, and P. Catarino, “On Leonardo Pisano dual quaternions,” J. Inf. Optim. Sci., vol. 45, no. 1, pp. 131–143 (2024).
[5] E. W. Dijkstra, “Archive: Fibonacci numbers and Leonardo numbers,” EWD 797, [Online]. Available: www.cs.utexas.edu. Jul. (1981).
[6] O. Diskaya, H. Menken, and P. Catarino, “On the hyperbolic Leonardo and hyperbolic François quaternions,” J. New Theory, vol. 42, pp. 74–85 (2023).
[7] C. Flaut and D. Savin, “Some special number sequences obtained from a difference equation of degree three,” Chaos Solitons Fractals, vol. 106, pp. 67–71 (2018).
[8] C. Flaut and D. Savin, “Some examples of division symbol algebras of degree 3 and 5,” Carpathian J. Math., vol. 31, no. 2, pp. 197–204 (2015).
[9] J. M. Grau, C. Miguel, and A. M. Oller-Marcen, “On the structure of quaternion rings over Z/nZ,” Adv. Appl. Clifford Algebras, vol. 25, no. 4, pp. 875–887 (2015).
[10] A. F. Horadam, “Complex Fibonacci numbers and Fibonacci quaternions,” Am. Math. Mon., vol. 70, pp. 289–291 (1963).
[11] S. Halici and A. Karatas, “On a generalization for quaternion sequences,” Chaos Solitons Fractals, vol. 98, pp. 178–182 (2017).
[12] E. G. Kocer, N. Tuglu, and A. Stakhov, “On the m-extension of the Fibonacci and Lucas p-numbers,” Chaos Solitons Fractals, vol. 40, pp. 1890–1906 (2009).
[13] K. Kuhapatanakul and J. Chobsorn, “On the generalized Leonardo numbers,” Integers, vol. 22, #A48 (2022).
[14] M. C. S. Mangueira, F. R. V. Alves, and P. M. M. C. Catarino, “Hybrid quaternions of Leonardo,” Trends Comput. Appl. Math., vol. 23, no. 1, pp. 51–62 (2022).
[15] C. J. Miguel and R. Serodio, “On the structure of quaternion rings over ℤp,” Int. J. Algebra, vol. 5, no. 27, pp. 1313–1325 (2011).
[16] D. Savin, “About special elements in quaternion algebras over finite fields,” Adv. Appl. Clifford Algebras, vol. 27, no. 2, pp. 1801–1813 (2017).
[17] D. Savin, “About division quaternion algebras and division symbol algebras,” Carpathian J. Math., vol. 32, no. 2, pp. 233–240 (2016).
[18] A. G. Shannon, “A note on generalized Leonardo numbers,” Notes Number Theory Discrete Math., vol. 25, no. 3, pp. 97–101 (2019).
[19] N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, The OEIS Foundation Inc., [Online]. Available: http://oeis.org.
[20] A. Stakhov and B. Rozin, “Theory of Binet formulas for Fibonacci and Lucas p-numbers,” Chaos Solitons Fractals, vol. 27, no. 5, pp. 1162–1177 (2006).
[21] E. Tan and H. H. Leung, “Some results on Horadam quaternions,” Chaos Solitons Fractals, vol. 138 (2020).
[22] E. Tan and H. H. Leung, “On Leonardo p-numbers,” Integers, vol. 23, #A7 (2023).
[23] E. Tan, D. Savin, and S. Yılmaz, “A new class of Leonardo hybrid numbers and some remarks on Leonardo quaternions over finite fields,” Mathematics, vol. 11, p. 4701 (2023).
[24] N. Tuglu, E. Gokcen Kocer, and A. Stakhov, “Bivariate Fibonacci-like p-polynomials,” Appl. Math. Comput., vol. 217, pp. 10239–10246 (2011).
[25] J. Zhong, J. Yao, and C.-L. Chung, “A note on incomplete Fibonacci-Lucas relations,” Symmetry, vol. 15, p. 2113 (2023).
