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.
Similar content being viewed by others
References
T. Amdeberhan, X. Chen, V. Moll and B. E. Sagan, “Generalized Fibonacci polynomials and Fibonomial coefficients,” Ann. Comb. 18 (4), 541–562 (2014).
U. Bednarz, D. Brod, I. Wloch and M. Wolowiec-Musial, “On three types of (2, k)-distance Fibonacci numbers and number decompositions,” Ars Combin. 118, 391–405 (2015).
U. Bednarz, A. Wloch and M. Wolowiec-Musial, “Distance Fibonacci numbers, their interpretations and matrix generators,” Comment. Math. 53, 35–46 (2013).
A. T. Benjamin, S. S. Plott, and J. A. Sellers, “Tiling proofs of recent sum identities involving Pell numbers,” Ann. Comb. 12 (3), 271–278 (2008).
R. da Silva, K. S. Oliveira and A. C. G. Neto, “On a four-parameter generalization of some special sequences,” Discr. Appl. Math. 243, 154–171 (2018).
S. Falcon, “Generalized \((k, r)\)-Fibonacci numbers,” General Math. Notes 25 (2), 148–158 (2014).
C. Flaut and V. Shpakivskyi, “On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions,” Adv. Appl. Clifford Algebr. 23 (3), 673–688 (2013).
J. H. Halton, “On the divisibility properties of Fibonacci numbers”, Fibonacci Quart. 4 (3), 217–240 (1966).
P. Haukkanen, “A note on Horadam’s sequence”, Fibonacci Quart. 40 (4), 358–361 (2002).
A. F. Horadam, “Basic Properties of Certain Generalized Sequence of Numbers”, Fibonacci Quart. 3 (3), 161–176 (1965).
E. Kiliç, “The Binet formula, sums and representations of generalized Fibonacci \(p\)-numbers,” Eur. J. Combin. 29, 701–711 (2008).
T. Koshy, Fibonacci and Lucas Numbers with Aplications (Wiley, New York, 2001).
A. Kreutz, J. Lelis, D. Marques and P. Trojovský, “The \(p\)-Adic order of the k-Fibonacci and k-Lucas numbers”, \(p\)-Adic Numb. Ultrametr. Anal. Appl. 9 (1), 15–21 (2017).
T. Lengyel, “The order of the Fibonacci and Lucas numbers”, Fibonacci Quart. 33 (3), 234–239 (1995).
I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers (John Wiley & Sons, 2008).
P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (Springer Science & Business Media, 2006).
C. Sanna, “The \(p\)-adic valuation of Lucas sequences”, Fibonacci Quart. 54 (2), 118–124 (2016).
N. Yilmaz and N. Taskara, “Matrix sequences in terms of Padovan and Perrin numbers”, J. Appl. Math. 2013 Article ID 941673 (2013).
Acknowledgments
The authors thank the anonymous referee for his/her helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046620040068