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.

中文翻译：

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一些特殊序列四参数推广的算术方法

摘要 在本文中，我们研究了最近引入的序列 $$F^{i}_{r,s}(k,n)$$ 的算术性质，其参数的某些值。这些新数字同时概括了许多众所周知的序列，包括斐波那契数列、佩尔数、雅各布斯塔尔数、帕多万数和纳拉亚纳数。我们将最近的斐波那契数的算术性质概括为 $$F^{1}_{r,s}(2,n)$$ 。此外，我们还研究了 $$2$$ -adic 顺序，并在此序列中找到了某些参数选择的阶乘。证明我们的结果所需的所有证明技术都是基本的。