Some Properties of the Generalized Leonardo Numbers

Year 2024, Issue: 47, 52 - 60, 30.06.2024

Yasemin Alp

https://doi.org/10.53570/jnt.1470097

Abstract

In this study, various properties of the generalized Leonardo numbers, which are one of the generalizations of Leonardo numbers, have been investigated. Additionally, some identities among the generalized Leonardo numbers have been obtained. Furthermore, some identities between Fibonacci numbers and generalized Leonardo numbers have been provided. In the last part of the study, binomial sums of generalized Leonardo numbers have been derived. The results obtained for generalized Leonardo numbers are reduced to Leonardo numbers.

Keywords

Binet, Fibonacci numbers, Leonardo numbers

References

• E. Lucas, Theorie des fonctions numeriques simplement periodiques, American Journal of Mathematics 1 (1878) 184-196.
• V. E. Hoggatt, Jr., Fibonacci and Lucas numbers, Fibonacci Association, Santa Clara, 1969.
• A. F. Horadam, A generalized Fibonacci sequence, American Mathematical Monthly 68 (1961) 455-459.
• D. Kalman, R. Mena, The Fibonacci numbers-exposed, Mathematics Magazine 76 (3) (2003) 167-181.
• T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley and Sons, New York, 2001.
• S. Vajda, Fibonacci and Lucas numbers and the golden section: Theory and applications, Halsted Press, Chichester, 1989.
• P. M. M. C. Catarino, A. Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2019) 75-86.
• Y. Alp, E. G. Kocer, Some properties of Leonardo numbers, Konuralp Journal Mathematics 9 (1) (2021) 183-189.
• U. Bednarz, M. Wołowiec-Musiał, Generalized Fibonacci–Leonardo numbers, Journal of Difference Equations and Applications 30 (1) (2024) 111-121.
• P. M. M. C. Catarino, A. Borges, A note on incomplete Leonardo numbers, Integers: Electronic Journal of Combinatorial Number Theory 20 (2020) 1-7.
• H. Gökbaş, A new family of number sequences: Leonardo-Alwyn numbers, Armenian Journal of Mathematics 15 (6) (2023) 1-13.
• S. Halıcı, S. Curuk, On the Leonardo sequence via Pascal-type triangles, Journal of Mathematics 2024 (2024) Article ID 9352986 8 pages.
• K. Kuhapatanakul, J. Chobsorn, On the generalized Leonardo numbers, Integers 22 (2022) #A48 7 pages.
• M. Kumari, K. Prasad, H. Mahato, P. M. M. C. Catarino, On the generalized Leonardo quaternions and associated spinors, Kragujevac Journal of Mathematics 50 (3) (2026) 425-438.
• K. Prasad, R. Mohanty, M. Kumari, H. Mahato, Some new families of generalized k-Leonardo and Gaussian Leonardo numbers, Communications in Combinatorics and Optimization 9 (2024) 539-553.
• A. G. Shannon, A note on generalized Leonardo numbers, Notes on Number Theory and Discrete Mathematics 25 (3) (2019) 97-101.
• A. G. Shannon, Ö. Deveci, A note on generalized and extended Leonardo sequences, Notes on Number Theory and Discrete Mathematics 28 (1) (2022) 109-114.
• M. Shattuck, Combinatorial proofs of identities for the generalized Leonardo numbers, Notes on Number Theory and Discrete Mathematics 28 (24) (2022) 778-790.
• E. Tan, H. H. Leung, On Leonardo p-numbers, Integers 23 (2023) #A7 11 pages.
• N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (2003), http://oeis.org, Accessed 24 Dec 2003.