Bidimensional extensions of cobalancing and Lucas-cobalancing numbers
Abstract
A new bidimensional version of cobalancing numbers and Lucasbalancing numbers are introduced. Some properties and identities satisfied by these new bidimensional sequences are studied.
Keywords
cobalancing numbers; Lucas-cobalancing numbers; balancing numbers; Lucas-balancing numbers; bidimensional recurrence relations
References
A. Behera and G.K. Panda, On the square roots of triangular numbers, Fibonacci Quart. 37 (1999), no. 2, 98–105.
F. Bezerra, F. Alves, and R. Vieira, Relações recorrentes bidimensionais e tridimensionais de Narayana, C.Q.D. – Revista Eletrónica Paulista de Matemática 18 (2020), Ediçaõ Iniciaçaõ Cientifica Julho, 12–28.
S. Bouroubi, On the square-triangular and balancing-numbers, Rostock. Math. Kolloq. 72 (2019/20), 73–80.
P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. (N.S.) 89 (2019), no. 1, 75–86.
P. Catarino, H. Campos, and P. Vasco, On some identities for balancing and cobalancing numbers, Ann. Math. Inform. 45 (2015), 11–24.
P. Catarino and P. Vasco, The generalized order-m(k-Pell) numbers, An. Stiinµ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 66 (2020), no. 1, 55–65.
J. Chimpanzo, P. Catarino, P. Vasco, and A. Borges, Bidimensional extensions of balancing and Lucas-balancing numbers, J. Discrete Math. Sci. Cryptogr. Accepted.
G.K. Gözeri, A. Özkoç, and A. Tekcan, Some algebraic relations on balancing numbers, Util. Math. 103 (2017), 217–236.
R. Keskin and O. Karaatlı, Some new properties of balancing numbers and square triangular numbers, J. Integer Seq. 15 (2012), no. 1, Article 12.1.4, 13 pp.
C. Kızılateş, A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos Solitons Fractals 130 (2020), 109449, 5 pp.
K. Liptai, Fibonacci balancing numbers, Fibonacci Quart. 42 (2004), no. 4, 330–340.
K. Liptai, Lucas balancing numbers, Acta Math. Univ. Ostrav. 14 (2006), no. 1, 43–47.
K. Liptai, F. Luca, Á. Pintér, and L. Szalay, Generalized balancing numbers, Indag. Math. (N.S.) 20 (2009), no. 1, 87–100.
P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Ann. Math. Inform. 37 (2010), 125–138.
R. Oliveira and F. Alves, Os números Gaussianos de Fibonacci e relações recorrentes bidimensionais, Revista Thema 16 (2019), no. 4, 745–754.
R. Oliveira, F. Alves, and R. Paiva, Identidades bi e tridimensionais para os números de Fibonacci na forma complexa, C.Q.D. – Revista Eletrónica Paulista de Matemática 11 (2017), Dezembro 2017, 91–106.
G.K. Panda and P.K. Ray, Cobalancing numbers and cobalancers, Int. J. Math. Math. Sci. 2005 (2005), no. 8, 1189–1200.
G.K. Panda and P.K. Ray, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bull. Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 1, 41–72.
P.K. Ray, Balancing and Cobalancing Numbers, Ph.D. Thesis, Department of Mathematics, National Institute of Technology, Rourkela, India, 2009.
P.K. Ray, On the properties of k-balancing numbers, Ain Shams Engineering Journal 9 (2018), no. 3, 395–402.
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. Available at OEIS
R. Vieira, F. Alves, and P. Catarino, Relações bidimensionais e identidades da sequência de Leonardo, Revista Sergipana de Matemática e Educação Matemática 4 (2019), no. 2, 156–173.
Department of Exact Sciences, Higher Polytechnic Institute of Soyo Angola
Department of Statistics, Mathematical Analysis and Optimization, University of Santiago de Compostela Spain
Department of Mathematics, University of Trás-os-Montes e Alto Douro Portugal
Department of Mathematics, University of Trás-os-Montes e Alto Douro Portugal
Department of Mathematics, University of Trás-os-Montes e Alto Douro Portugal
https://orcid.org/0000-0001-6917-5093
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
