Milena Carolina dos Santos Mangueira, Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 101–111
DOI: 10.7546/nntdm.2021.27.1.101111
Full paper (PDF, 230 Kb)
Download Corrigendum: PDF, 140 Kb)
Details
Authors and affiliations
Milena Carolina dos Santos Mangueira
Department of Mathematics, Federal Institute of Education, Science and Technology
of State of Ceara – IFCE, Treze of Maio, Brazil
Renata Passos Machado Vieira
Department of Mathematics, Federal Institute of Education, Science and Technology
of State of Ceara – IFCE, Treze of Maio, Brazil
Francisco Regis Vieira Alves
Department of Mathematics, Federal Institute of Education, Science and Technology
of State of Ceara – IFCE, Treze of Maio, Brazil
Paula Maria Machado Cruz Catarino
University of TrasosMontes and Alto Douro – UTAD
Vila Real, Portugal
Abstract
This article deals with the generalization of the matrix form of the Oresme sequence, extending it to the field of integers. In addition, the two valid generating matrices were discussed, through the permutation of the rows and columns, respectively, totaling two valid matrices of Oresme for the lefthand side, and two more generating matrices of Oresme for the righthand side. In addition, a new Oresme sequence was introduced, given through the hybridization process of these numbers, obtaining mathematical properties and theorems, inherent to this process.
Keywords
 Matrix form of Oresme
 Oresme’s hybrid numbers
 Oresme sequence
2010 Mathematics Subject Classification

 11B37
 11B39
 11B83
References
 Alves, F. R. V. (2019). Sequencia de Oresme e algumas propriedades (matriciais) generalizadas. C.Q.D.– Revista Eletronica Paulista de Matematica, 16, 28–52.
 Bilgici, G., Tokeser, U., & Unal, Z. (2017). kFibonacci and kLucas Generalized
Quaternions. Konuralp Journal of Mathematics, 5(2), 102–113.  Catarino, P. (2019). On kPell hybrid numbers. Journal of Discrete Mathematical Sciences and Cryptography, 22(1), 83–89.
 Carvalho, C. (2019). Numeros hıbridos e sua visualizacao no geogebra. Dissertacao (Mestrado Profissional) Universidade Estadual do Ceara, Centro de Ciencia e Tecnologia, Mestrado Profissional em Matematica em Rede Nacional, Fortaleza.
 CerdaMorales, G. (2018). Investigation of generalized hybrid Fibonacci numbers and their properties. arXiv preprint, arXiv:1806.02231.
 Cook, C. (2004). Some sums related to sums of Oresme numbers. In: Howard F. T. (eds) Applications of Fibonacci Numbers, Springer, Dordrecht, 87–99.
 CerdaMorales, G. (2019). Oresme Polynomials and Their Derivatives. arXiv preprint arXiv:1904.01165.
 Dasdemir, A., & Bilgici, G. (2019). Gaussian Mersenne numbers and generalized Mersenne quaternions. Notes on Number Theory and Discrete Mathematics, 25(3), 87–96.
 Dasdemir, A. (2016). Generalizations of Modified Pell and Pell Lucas Sequences and Their Generating Matrices and Some Sums. Erzincan University Journal of Science and Technology, 9(3), 178–184.
 Ercolano, J. (1979). Matrix generators of Pell sequences. The Fibonacci Quarterly, 17(1), 71–77.
 Horadam, A. F. (1974). Oresme numbers. The Fibonacci Quarterly, 12(3), 267–271.
 Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. [S.l.]: New York: Wiley and Sons publications, 1.
 Mangueira, M. C. dos S., et al. (2020). The Hybrid Numbers of Padovan and Some Identities. Annales Mathematicae Silesianae, 34(2), 256–267.
 Mangueira, M. C. dos S., & Alves, F. R. V. (2020). Numeros hıbridos de Fibonacci e Pell. Revista Thema, 17(3), 831–842.
 Mangueira, M. C. dos S., Alves, F. R. V., & Catarino, P. M. M. (2020). Numeros hıbridos de Mersenne. C.Q.D. – Revista Eletronica Paulista de Matematica, 18, 1–11.
 Ozdemir, M. (2018). Introduction to hybrid numbers. Advances in Applied Clifford Algebras, 28, Art. No. 11.
 Senturk, T. D., Bilgici, G., Dasdemir, A., & Unal, Z. (2020). A study on Horadam hybrid numbers. Turkish Journal of Mathematics, 44, 1212–1221.
 Senturk, T. D., Dasdemir, A., Bilgici, G., & Unal, Z. (2019). On unrestricted Horadam generalized quaternions. Utilitas Mathematica, 110, 89–98.
 SzynalLiana, A. (2018). The Horadam hybrid numbers. Discussiones Mathematicae – General Algebra and Applications, Sciendo, 38(1), 91–98.
 SzynalLiana, A., & Wloch, I. (2019). On Jacobsthal and Jacobsthal–Lucas hybrid numbers. Annales Mathematicae Silesianae, Sciendo, 33(1), 276–283.
 Vieira, R., Alves, F. R., & Catarino, P. (2020). (s1; s2)Padovan matrix sequence and the case of generalization. Analele stiintifice ale Universitatii “Al. I. Cuza” din Iasi Mat. (N.S.) (to appear).
Corrigendum
 Mangueira, M. C. dos S., Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. (2021). Corrigendum to “The Oresme sequence: The generalization of its matrix form and its hybridization process” [Notes on Number Theory and Discrete Mathematics, Vol. 27, 2021, No. 1, 101–111]. Notes on Number Theory and Discrete Mathematics, 27(4), 276279.
Related papers
Cite this paper
Mangueira, M. C. dos S., Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. (2021). The Oresme sequence: The generalization of its matrix form and its hybridization process. Notes on Number Theory and Discrete Mathematics, 27(1), 101111, DOI: 10.7546/nntdm.2021.27.1.101111.