An Arithmetic Approach to a Four-Parameter Generalization of Some Special Sequences
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RESEARCH ARTICLES
An Arithmetic Approach to a Four-Parameter Generalization of Some Special Sequences∗ R. da Silva1** , A. C. da Graca ¸ Neto2*** , and K. S. de Oliveira3**** 1
Universidade Federal de Sao ˜ Paulo, Sao ˜ Jose´ dos Campos - SP, 12247-014, Brazil 2
Universidade do Estado do Amazonas, Manaus - AM, 69055-038, Brazil 3 Universidade Federal do Amazonas, Manaus - AM, 69103-128, Brazil
Received June 6, 2020; in final form, September 22, 2020; accepted September 25, 2020
Abstract—In this paper, we study arithmetic properties of the recently introduced sequence i Fr,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 1 (2, n). In numbers. We generalize a recent arithmetic property of the Fibonacci numbers to Fr,s 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. DOI: 10.1134/S2070046620040068 Key words: generalized Fibonacci number, Pell numbers, Padovan numbers, Jacobsthal numbers, Narayana numbers, arithmetic properties.
1. INTRODUCTION i (k, n) defined in the following way: given integers i, k, r, s ≥ 1, In [5] we introduced the sequence Fr,s and n ≥ −1, ⎧ ⎪ i i ⎪ ⎪ ⎨ rFr,s (k, n − i) + sFr,s (k, n − k), for n ≥ max{k, i} − 1, i (k, n) = Fr,s
r i , for i ≤ k and n = −1, 0, 1, . . . , k − 2, ⎪ ⎪ ⎪ ⎩ s n+1 k , for k < i and n = −1, 0, 1, . . . , i − 2. n+1
(1.1)
This sequence, as pointed out in [5], simultaneously generalizes a number of well-known integer sequences, including the Fibonacci (Fn ), Pell (Pn ), Jacobsthal (Jn ), Padovan (Pv (n)), and Narayana (un ) numbers. Table 1 of [5] presents a list of some sequences that are generalized by the numbers i (k, n), including some numbers studied in [1–4, 6, 7, 11]. Fr,s i (k, n) also has a nice combinatorial In addition to generalizing many well-known sequences, Fr,s interpretation in terms of tilings (see [5, Theorem 3.1]), which allowed us to prove many interesting identities. As a consequence, many new identities involving the well-known numbers generalized by i (k, n) were obtained. Fr,s i (k, n) for some specific values of the In this paper, we are interested in the arithmetic aspects of Fr,s 1 parameters i, r, and s. We obtain an arithmetic properties of Fr,s (2, n), which generalizes a recent result involving the Fibonacci numbers. As a consequence, we derive a similar result for the Pell numbers. We ∗
The text was submitted by the authors in English. E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] **
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AN ARITHMETIC APPROACH
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i (3, n), for i = 1, 2, and we present the solutions for the Diophantine also study the 2-adic order of F1,1 i (3, n) = m!, for i = 1, 2. equation F1,1 1 (2, n). The This paper is organized as follows. In Section 2, we present arithmetic properties of Fr,s i (3, n), for i = 1, 2, is the c
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