On digit expansions with respect to linear recurrences

Theoretical Computer Science, 2004

We exhibit and study various regularity properties of the sequence (R(n)) n 1 which counts the number of different representations of the positive integer n in the Fibonacci numeration system. The regularity properties in question are observed by representing the sequence as a two-dimensional array consisting of an infinite number of rows L 1 , L 2 , L 3 , . . . where each L k contains f k−1 (the k − 1st Fibonacci number) entries of the sequence (R(n)). We give a purely combinatorial recursive algorithm for generating each row L k from previous rows L j with j < k. We then show that for each positive integer m, and for all k 2m, the number of occurrences of m in L k is a constant rk(m) depending only on m. The function rk(m) has many interesting number theoretic properties and is intimately connected to the Euler -function.

Applied Mathematical Sciences, 2015

We provide a formula for the n th term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation x+x −k = 2. We then extend our results to k-generalized Horadam (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for k-periodic k-nary Fibonacci sequence can also be derived.

Mathematics, 2021

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

arXiv (Cornell University), 2021

Let Fn(k) be the generalized Fibonacci number defined by (with Fi(k) abbreviated to Fi): Fn = Fn−1 + Fn−2 + • • • + F n−k , for n ≥ k, and the initial values (F0, F1, ..., F k−1). Let Bn(k, j) be Fn(k) with initial values given by Fj = 1 and, for i < j and j < i < k, Fi = 0. This paper shows that any Fn(k) can be expressed as the sum of Bn(k, j)s. This paper also expresses Bn(k, j) and Fn(k) as finite sums, derives some properties and evaluates their 2-adic order for a range of values of k, j and n and those of Bn(3, j) and Bn(4, j) for most values of j and n.

2018

We provide a formula for the nth term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation x+x−k = 2. We then extend our results to k-generalized Horadam (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for k-periodic k-nary Fibonacci sequence can also be derived. Mathematics Subject Classification: 11B39, 11B50.

2010

Journal of Integer Sequences, 2007

In this note, we study Fibonacci-like sequences that are defined by the recurrence Sk = a, Sk+1 = b, Sn+2 ≡ Sn+1 + Sn (mod n + 2) for all n ≥ k, where k, a, b ∈ N, 0 ≤ a &lt; k, 0 ≤ b &lt; k + 1, and (a, b) 6= (0, 0). We will show that the number α = 0.SkSk+1Sk+2 · · · is irrational. We also propose a conjecture on the pattern of the sequence {Sn}n≥k.

arXiv (Cornell University), 2024

One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer recurrence sequences as rational polynomial linear combinations of Fibonacci numbers.

Journal of Number Theory, 1997

We will show that the sum-of-digits function has an asymptotic Gaussian behaviour, and we deduce some new summational formulae. More precisely we consider numeration systems relative to substitutive bases, whose general case contains among others the case of Parry linear recurrent bases (as the Fibonacci sequence for instance).

2016

In this article, we introduce a new generalization of Fibonacci sequence and we call it as k-Fibonacci-Like sequence. After that we obtain some fundamental properties for k-Fibonacci-Like sequence and also we present some relations among k-Fibonacci-Like sequence, k-Fibonacci sequence and k-Lucas sequence by some algebraic methods. MSC: 11B37, 11B39.