Notes on a General Sequence

Reza Farhadian 1  and Rafael Jakimczuk 2
  • 1 Department of Statistics, Lorestan University, Iran, Khorramabad
  • 2 División Matemática, Universidad Nacional de Luján, Luján, Buenos Aires, República, Argentina


Let {rn}n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn, where α, β are real numbers with α > 0 and β > 1. In this note we prove some limits that connect this sequence to the number e. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.

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  • [1] M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972.

  • [2] G. Bilgici and T.D. Şentürk, Some addition formulas for Fibonacci, Pell and Jacobsthal numbers, Ann. Math. Sil. 33 (2019), 55–65.

  • [3] R. Jakimczuk, Functions of slow increase and integer sequences, J. Integer Seq. 13 (2010), Article 10.1.1, 14 pp.

  • [4] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.

  • [5] T. Koshy, Sums of Fibonacci–Pell–Jacobsthal products, Internat. J. Math. Ed. Sci. Tech. 44 (2013), 559–568.

  • [6] B. Pritsker, The Equations World, Dover Publications, New York, 2019.

  • [7] J. Rey Pastor, P. Pi Calleja and C.A. Trejo, Análisis Matemático, Vol. 1, Editorial Kapelusz, Buenos Aires, 1969.

  • [8] A.P. Stakhov, The golden section in the measurement theory, Comput. Math. Appl. 17 (1989), 613–638.

  • [9] A. Szynal-Liana and I. Włoch, On Jacobsthal and Jacobsthal-Lucas hybrid numbers, Ann. Math. Sil. 33 (2019), 276–283.


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