An Arithmetic Approach to a Four-Parameter Generalization of Some Special Sequences

Abstract

In this paper, we study arithmetic properties of the recently introduced sequence \(F^{i}_{r,s}(k,n)\), for some values of its parameters. These new numbers simultaneously generalizes a number of well-known sequences, including the Fibonacci, Pell, Jacobsthal, Padovan, and Narayana numbers. We generalize a recent arithmetic property of the Fibonacci numbers to \(F^{1}_{r,s}(2,n)\). In addition, we also study the \(2\)-adic order and find factorials in this sequence for certain choices of the parameters. All the proof techniques required to prove our results are elementary.

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Acknowledgments

The authors thank the anonymous referee for his/her helpful comments and suggestions.

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Correspondence to R. da Silva.

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da Silva, R., da Graça Neto, A.C. & de Oliveira, K.S. An Arithmetic Approach to a Four-Parameter Generalization of Some Special Sequences. P-Adic Num Ultrametr Anal Appl 12, 322–332 (2020). https://doi.org/10.1134/S2070046620040068

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Keywords

  • generalized Fibonacci number
  • Pell numbers
  • Padovan numbers
  • Jacobsthal numbers
  • Narayana numbers
  • arithmetic properties