A Note on Bi-Periodic Leonardo Sequence

Paula Maria Machado Cruz Catarino; Elen Spreafico

doi:10.52737/18291163-2024.16.5-1-17

Authors

Paula Catarino University of Trás-os-Montes e Alto Douro
Elen Spreafico Universidade Federal de Mato Grosso do Sul

DOI:

https://doi.org/10.52737/18291163-2024.16.5-1-17

Keywords:

Leonardo sequence, bi-periodic Fibonacci sequence, Binet's formula, generating function, Catalan's identity, Cassini's identity, d'Ocgane's identity

Abstract

In this work, we define a new generalization of the Leonardo sequence by the recurrence relation $GLe_n=aGLe_{n-1}+GLe_{n-2}+a$ (for even $n$) and $GLe_n=bGLe_{n-1}+GLe_{n-2}+b$ (for odd $n$) with the initial conditions $GLe_0=2a-1$ and $GLe_1=2ab-1$, where $a$ and $b$ are real nonzero numbers. Some algebraic properties of the sequence $\{GLe_n\}_{n \geq 0}$ are studied and several identities, including the generating function and Binet's formula, are established.

References

Y. Alp and E.G. Koçer, Some properties of Leonardo numbers. Konuralp J. Math., 9 (2021), no. 1, pp. 183-189.

F.R.V. Alves and R.P.M. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab. Int. Electron. J. Math. Educ., 15 (2020), no. 2, Article No: em0575, pp 1-9. https://doi.org/10.29333/iejme/6440 DOI: https://doi.org/10.29333/iejme/6440

G. Bilgici, Two generalizations of Lucas sequence. Appl. Math. Comput., 245 (2014), pp 526-538. https://doi.org/10.1016/j.amc.2014.07.111 DOI: https://doi.org/10.1016/j.amc.2014.07.111

P. Catarino and A. Borges, A note on incomplete Leonardo numbers. Integers, 20 (2020), no. A43, pp 1-7.

P. Catarino and A. Borges, On Leonardo numbers. Acta Math. Univ. Comen., 89 (2019), no. 1, pp 75-86.

M. Edson and O. Yayenie, A new generalization of Fibonacci sequence and extended Binet's formula. Integers, 9 (2009), no. 6, pp 639-654. https://doi.org/10.1515/integ.2009.051 DOI: https://doi.org/10.1515/INTEG.2009.051

H. Gokbas, A new family of number sequences: Leonardo-Alwyn numbers. Armen. J. Math., 15 (2023), no. 6, pp 1-13. DOI: https://doi.org/10.52737/18291163-2023.15.6-1-13

N. Kara and F. Yilmaz, On hybrid numbers with Gaussian Leonardo coefficients. Mathematics, 11 (2023), no. 6, pp 1--12. https://doi.org/10.3390/math11061551 DOI: https://doi.org/10.3390/math11061551

T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, New York, 2014. https://doi.org/10.1007/978-1-4614-8489-9_7 DOI: https://doi.org/10.1007/978-1-4614-8489-9

T. Koshy, Fibonacci and Lucas numbers with applications, Vol. 1, Springer, John Wiley and Sons, New Jersey, 2018.

T. Koshy, Fibonacci and Lucas numbers with applications, Vol. 2, Springer, John Wiley and Sons, New Jersey, 2019. DOI: https://doi.org/10.1002/9781118742297

K. Kuhapatanakul and J. Chobsorn, On the generalized Leonardo numbers. Integers 22 (2022), no. A48, pp 1-7.

M.C.S. Mangueira, F.R.V. Alves and P.M.M.C. Catarino, Os números híbridos de K-Leonardo. Brazilian Electronic Journal of Mathematics, 3 (2022), no.5, pp 71-84. https://doi.org/10.14393/bejom-v3-n5-2022-61534 DOI: https://doi.org/10.14393/BEJOM-v3-n5-2022-61534

E. Tan and H. Leung, On Leonardo p-numbers. Integers, 23 (2023), no. A7, pp 1-11. DOI: https://doi.org/10.3390/math11224701

S. Uygun and E. Owusu, A new generalization of Jacobsthal Lucas numbers (bi-periodic Jacobsthal Lucas sequence). J. Adv. Math. Comput. Sci., 34 (2019), no. 5, pp 1-13. https://doi.org/10.9734/jamcs/2019/v34i530226 DOI: https://doi.org/10.9734/jamcs/2019/v34i530226

S. Uygun and E. Owusu, A new generalization of Jacobsthal numbers (Bi-Periodic Jacobsthal Sequences). J. Math. Anal., 7 (2016), no.5, pp 28-39.