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.