Abstract
The main purpose of this paper is to study some properties of Vietoris’ number sequence and present some techniques, using special types of matrices that generates this number sequence.
Similar content being viewed by others
References
Aggarwal, D., Joux, A., Prakash, A., Santha, M.: A New Public-Key Cryptosystem via Mersenne Numbers. In: Shacham, H., Boldyreva, A. (eds.) Advances in Cryptology-CRYPTO 2018. Lecture Notes in Computer Science, vol. 10993. Springer, Cham (2018)
Arslan, S., Köken, F.: The Jacobsthal and Jacobsthal–Lucas numbers via square roots of matrices. Int. Math. Forum 11(11), 513–520 (2016)
Arslan, S., Köken, F.: The Pell and Pell–Lucas numbers via square roots of matrices. J. Inform. Math. Sci. 8(3), 159–166 (2016)
Askey, R., Steinig, J.: Some positive trigonometric sums. Trans. AMS 187(1), 295–307 (1974)
Beunardeau, M., Connolly, A., Géraud, R., Naccache, D.: On the hardness of the Mersenne Low Hamming Ratio assumption. Tech. Rep. Cryptol. ePrint Archive, 2017/522 (2017)
Cação, I., Falcão, M.I., Malonek, H.R.: Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory 12(2), 371–391 (2012)
Cação, I., Falcão, M.I., Malonek, H.R.: Hypercomplex polynomials, Vietoris’ rational numbers and a related integer numbers sequence. Complex Anal. Oper. Theory 11(5), 1059–1076 (2017)
Cação, I., Falcão, M.I., Malonek, H.R.: On generalized Vietoris’ number sequences. Discrete Appl. Math. 269, 77–85 (2019)
Cahill, N.D., D’Errico, J.R., Narayan, D.A., Narayan, J.Y.: Fibonacci determinants. Coll. Math. J. 33(3), 221–225 (2002)
Catarino, P., Campos, H., Vasco, P.: On the Mersenne sequence. Annales Mathematicae et Informaticae 46, 37–53 (2016)
Cerin, Z.: On factors of sums of consecutive Fibonacci and Lucas numbers. Annales Mathematicae et Informaticae 41, 19–25 (2013)
Cook, C.K., Bacon, M.R.: Some identities for Jacobsthal and Jacobsthal–Lucas numbers satisfying higher order recurrence relations. Annales Mathematicae et Informaticae 41, 27–39 (2013)
Falcon, S.: On the generating matrices of the k-Fibonacci numbers. Proyecciones 32(4), 347–357 (2013)
Faye, B., Luca, L.: Pell and Pell–Lucas numbers with only one distinct digit. Annales Mathematicae et Informaticae 45, 55–60 (2015)
Kilic, E., Tasci, D., Haukkanen, P.: On the generalized Lucas sequences by Hessenberg matrices. Ars Combinat. 95, 383–395 (2010)
Koshy, T.: Fibonacci and Lucas Numbers with Applications. Volume II, First Edition, Wiley, (2019)
Koshy, T.: Catalan Numbers with Applications. Oxford University Press Inc, New York (2009)
Koshy, T.: Fibonacci and Lucas Numbers with Applications, vol. I, 2nd edn. Wiley, Hoboken (2018)
Koshy, T., Gao, Z.: Catalan numbers with Mersenne subscripts. Math. Sci. 38(2), 86–91 (2013)
Kurosawa, T., Tachiya, Y., Tanaka, T.: Algebraic relations with the infinite products generated by Fibonacci numbers. Annales Mathematicae et Informaticae 41, 107–119 (2013)
Onphaeng, K., Pongsriiam, P.: Jacobsthal and Jacobsthal–Lucas numbers and sums introduced by Jacobsthal and Tverberg. J. Integer Seq. 20 (2017). (Article 17.3.6)
Qi, F., Guo, B.-N.: Integral representations of Catalan numbers and their applications. Mathematics 5(3), 40 (2017). https://doi.org/10.3390/math5030040
Qia, F., Shic, X.-T., Liud, F.-F.: An integral representation, complete monotonicity, and inequalities of the catalan numbers. Filomat 32(2), 575–587 (2018)
Ruscheweyh, S.T., Salinas, L.: Stable functions and Vietoris’ theorem. J. Math. Anal. Appl. 291, 596–604 (2004)
Sloane, N.J.A., Plouffe, S.: The Encyclopedia of Integer Sequences. Academic Press, San Diego (1995)
Sun, Z.: Binomial coefficients, Catalan numbers and Lucas quotients. Sci. China Math. 53, 2473–2488 (2010)
Acknowledgements
The research of the authors was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020. Also the first author thanks the support given by the Project UID/CED/00194/2020.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the authors was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020. Also the first author thanks the support given by the Project UID/CED/00194/2020.
Rights and permissions
About this article
Cite this article
Catarino, P., Almeida, R. A Note on Vietoris’ Number Sequence. Mediterr. J. Math. 19, 41 (2022). https://doi.org/10.1007/s00009-021-01952-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-021-01952-w