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A Note on Vietoris’ Number Sequence

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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.

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References

  1. 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)

    Google Scholar 

  2. Arslan, S., Köken, F.: The Jacobsthal and Jacobsthal–Lucas numbers via square roots of matrices. Int. Math. Forum 11(11), 513–520 (2016)

    Article  Google Scholar 

  3. Arslan, S., Köken, F.: The Pell and Pell–Lucas numbers via square roots of matrices. J. Inform. Math. Sci. 8(3), 159–166 (2016)

    Google Scholar 

  4. Askey, R., Steinig, J.: Some positive trigonometric sums. Trans. AMS 187(1), 295–307 (1974)

    Article  MathSciNet  Google Scholar 

  5. 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)

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. Cação, I., Falcão, M.I., Malonek, H.R.: On generalized Vietoris’ number sequences. Discrete Appl. Math. 269, 77–85 (2019)

    Article  MathSciNet  Google Scholar 

  9. Cahill, N.D., D’Errico, J.R., Narayan, D.A., Narayan, J.Y.: Fibonacci determinants. Coll. Math. J. 33(3), 221–225 (2002)

    Article  Google Scholar 

  10. Catarino, P., Campos, H., Vasco, P.: On the Mersenne sequence. Annales Mathematicae et Informaticae 46, 37–53 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Cerin, Z.: On factors of sums of consecutive Fibonacci and Lucas numbers. Annales Mathematicae et Informaticae 41, 19–25 (2013)

    MathSciNet  MATH  Google Scholar 

  12. 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)

    MathSciNet  MATH  Google Scholar 

  13. Falcon, S.: On the generating matrices of the k-Fibonacci numbers. Proyecciones 32(4), 347–357 (2013)

    Article  MathSciNet  Google Scholar 

  14. Faye, B., Luca, L.: Pell and Pell–Lucas numbers with only one distinct digit. Annales Mathematicae et Informaticae 45, 55–60 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Kilic, E., Tasci, D., Haukkanen, P.: On the generalized Lucas sequences by Hessenberg matrices. Ars Combinat. 95, 383–395 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Koshy, T.: Fibonacci and Lucas Numbers with Applications. Volume II, First Edition, Wiley, (2019)

  17. Koshy, T.: Catalan Numbers with Applications. Oxford University Press Inc, New York (2009)

    MATH  Google Scholar 

  18. Koshy, T.: Fibonacci and Lucas Numbers with Applications, vol. I, 2nd edn. Wiley, Hoboken (2018)

    Book  Google Scholar 

  19. Koshy, T., Gao, Z.: Catalan numbers with Mersenne subscripts. Math. Sci. 38(2), 86–91 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Kurosawa, T., Tachiya, Y., Tanaka, T.: Algebraic relations with the infinite products generated by Fibonacci numbers. Annales Mathematicae et Informaticae 41, 107–119 (2013)

    MathSciNet  MATH  Google Scholar 

  21. 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)

  22. Qi, F., Guo, B.-N.: Integral representations of Catalan numbers and their applications. Mathematics 5(3), 40 (2017). https://doi.org/10.3390/math5030040

    Article  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. Ruscheweyh, S.T., Salinas, L.: Stable functions and Vietoris’ theorem. J. Math. Anal. Appl. 291, 596–604 (2004)

    Article  MathSciNet  Google Scholar 

  25. Sloane, N.J.A., Plouffe, S.: The Encyclopedia of Integer Sequences. Academic Press, San Diego (1995)

    MATH  Google Scholar 

  26. Sun, Z.: Binomial coefficients, Catalan numbers and Lucas quotients. Sci. China Math. 53, 2473–2488 (2010)

    Article  MathSciNet  Google Scholar 

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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.

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Correspondence to P. Catarino.

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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.

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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

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  • DOI: https://doi.org/10.1007/s00009-021-01952-w

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