axioms Article On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic; [email protected]; Tel.: +42-049-333-2860 (r,a) Abstract: For r ≥ 2 and a ≥ 1 integers, let (tn )n≥1 be the sequence of the (r, a)-generalized (r,a) (r,a) (r,a) Fibonacci numbers which is defined by the recurrence tn = tn−1 + ··· + tn−r for n > r, with initial (r,a) (r,a) values ti = 1, for all i 2 [1, r − 1] and tr = a. In this paper, we shall prove (in particular) that, for any given r ≥ 2, there exists a positive proportion of positive integers which can not be written as (r,a) tn for any (n, a) 2 Z≥r+2 × Z≥1. Keywords: natural density; fibonacci r-numbers; recurrence sequences 1. Introduction We start by recalling that if A is a set of positive integers, the natural density of A, denoted by d(A), is the following limit (if it exists) #A(x) d(A) := lim , x!¥ x Citation: Trojovský, P. On the Natural Density of Sets Related to where, A(x) := A\ [1, x], for x > 0 (also lim infx!¥ #A(x)/x and lim supx!¥ #A(x)/x Generalized Fibonacci Numbers of are called lower density and upper density, respectively). For example, if F := (pFn)n≥0 Order r. Axioms 2021, 10, 144. is the Fibonacci sequence, one has that F(x) ≤ (log x)/(log f), where f = (1 + 5)/2. https://doi.org/10.3390/ In particular, the natural density of Fibonacci numbers is zero and, so, almost all positive axioms10030144 integers are non-Fibonacci numbers (i.e., d(Z>0nF) = 1). Clearly, this is not a surprising fact, given the exponential nature of Fibonacci sequence. We refer the reader to the classical Academic Editor: Gradimir V. work of Niven [1] (and references therein) for more details about natural (and asymptotic) Milovanovi´c density (see also books [2,3] and papers [4–8] for more recent results). It is especially interesting that the some kind of “combinations” of zero density sets Received: 11 June 2021 may have positive density. For instance, the set of powers of two and the set of prime Accepted: 30 June 2021 numbers have zero density but, in 1934, a classical result of Romanov [9] implied that the Published: 1 July 2021 set of positive integers which are not of the form p + 2k, for some p prime and k ≥ 0, has upper density smaller than one. Publisher’s Note: MDPI stays neutral It is also possible to study the density of a set in some prescribed subset of Z≥1. More with regard to jurisdictional claims in precisely, let A and B be elements of P( ≥1) (the power set of ≥1, i.e., the set of all subsets published maps and institutional affil- Z Z of positive integers), we name d (A) the B-density of A, as the limit (if it exists) iations. B #AB(x) dB(A) := lim , x!¥ x where, AB(x) := fm ≤ x : bm 2 Ag (for x > 0) and B := fb1, b2, ...g (with bi < bi+1). Copyright: © 2021 by the author. (A) = P( ) ! [ ] By convention dÆ 0 and note that dZ≥1 : Z≥1 0, 1 is the standard natural density. Licensee MDPI, Basel, Switzerland. As any very well-studied object in mathematics, the Fibonacci sequence possesses This article is an open access article many kinds of generalizations (see, e.g., [10–14]). One of the most well-known generaliza- distributed under the terms and (r) conditions of the Creative Commons tion is probably the sequence of generalized Fibonacci numbers of order r, denoted by (tn )n≥0, Attribution (CC BY) license (https:// which is defined by the rth order recurrence creativecommons.org/licenses/by/ (r) (r) (r) 4.0/). tn = tn−1 + ··· + tn−r Axioms 2021, 10, 144. https://doi.org/10.3390/axioms10030144 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 144 2 of 6 (r) (r) with initial conditions t0 = 0 and ti = 1, for i 2 [1, r − 1]. For r = 2, we have the sequence of Fibonacci numbers, for r = 3, we have the Tribonacci numbers and so on. For recent results on this sequence, we cite [15] and its annotated bibliography. Here we are interested in a related generalization. More precisely, let r ≥ 2 and a ≥ 0 (r,a) be integers. The (r, a)-generalized Fibonacci sequence (tn )n≥1 is defined by 8 >1, if 1 ≤ n ≤ r − 1; (r,a) < tn = a, if n = r; :> r (r,a) ∑i=1 tn−i , if n ≥ r + 1. (r) (r,r−1) Note that (tn )n≥1 = (tn )n≥1. For our purpose, we consider the previous se- (r,a) quence only after its (r + 1)th term, i.e., we denote Tr,a := (tn )n>r+1. As before, by its exponential nature, it holds that d(Tr,a) = 0. Now, we turn or attention to the a specific “combination” of these sets, namely, their union. Thus, the following question arises: Are there infinitely many positive integers which do not belong to Tr = [a≥1Tr,a? If so, does this “exception set” represent a positive proportion (i.e., with positive natural density) of the positive integers? In this paper, we answer (positively) this question by proving a more general result. More precisely, Theorem 1. Let r ≥ 2 be an integer. Then there exists B 2 P(Z≥1) (depending only on r) with 1 d(B) = 2r−1(2r − 1) such that 1 (T ) < dB r r , (1) ar(ar − 1) r r−1 where ar is the only positive real root of x − x − · · · − x − 1. In particular, for any r ≥ 2, there exists a positive proportion of positive integers which do not belong to Tr,a, for all a 2 Z≥1. Remark 1. Table1 shows that the upper bound for dB(Tr) provided in the Theorem1, say ur, decreases significantly as r grows. Table 1. The upper bound for dB(Tr). r 2 3 4 5 6 7 8 9 10 ur 0.6180 0.1914 0.0781 0.0352 0.0166 0.0081 0.0039 0.0019 0.0009 We organize this paper as follows. In Section2, we will present some helpful properties (r,a) of the sequence (tn )n. The third section is devoted to the proof of Theorem1. 2. Auxiliary Results Before proceeding further, we shall present some useful tools related to the previ- ous sequences. (r,a) r r−1 The characteristic polynomial of the sequence (tn )n is yr(x) = x − x − · · · − x − 1 which has only one root outside the unit circle, say ar, which is located in the interval (2(1 − 2−r), 2) (see [16]). Throughout this work, in order to simplify the notations, we shall write a for ar and for integers a < b, we write [a, b] for fa, a + 1, . , bg. (r) In 2018, Young [17] (p. 3) found a closed formula for tn in the range n 2 [r + 1, 2r], namely, (r) i−1 tr+i = 2 (2r − 3) + 1, Axioms 2021, 10, 144 3 of 6 (r,a) for all i 2 [1, r]. The next result generalizes this fact for (tn )n: Lemma 1. The identity (r,a) i−1 tr+i = 2 (r + a − 2) + 1 holds for all i 2 [1, r]. Proof. To prove this identity, we shall use (finite) induction on i 2 [1, r]. For the basis case (r,a) i = 1, since tr = a, one has that (r,a) 1−1 t + = a + 1 + ··· + 1 = a + r − 1 = 2 (r + a − 2) + 1. r 1 | {z } r−1 (r,a) i−1 Suppose (by the induction hypothesis) that tr+i = 2 (r + a − 2) + 1, for some i 2 [1, r − 1]. Then (r,a) (r,a) (r,a) (r,a) (r,a) (r,a) tr+i+1 = tr+i + tr+i−1 + ··· + ti+1 = 2tr+i − ti = 2 · (2i−1(r + a − 2)) − 1 = 2i(r + a − 2) + 1 which completes the induction process. Before stating the next lemma, we recall that the sequence of k-bonacci numbers (or (r) k-generalized Fibonacci numbers) (Fn )n≥−(r−2), is the kth order linear recurrence which (r,a) satisfies the same recurrence as (tn )n≥1, namely, (r) (r) (r) Fn = Fn−1 + ··· + Fn−r, (r) but with r initial values 0, . , 0, 1 = F1 (see, e.g., [18,19]). (r,a) (r) The third lemma relates the sequences (tn )n and (Fn )n. Specifically, we have Lemma 2. If n ≥ r, then (r,a) (r) (r,0) tn = aFn−r+1 + tn . Proof. Define the sequence (Xn)n≥r, by (r) (r,0) Xn := aFn−r+1 + tn . (r,a) Therefore, we want to prove that Xn = tn , for all n ≥ r. For that, first observe that (r) (r,0) Xn = aFn−r+1 + tn (r) (r,0) Xn+1 = aFn−r+2 + tn+1 (r) (r,0) Xn+2 = aFn−r+3 + tn+2 . (r) (r,0) Xn+r−1 = aFn + tn+r−1. By summing up all previous equalities, we obtain that Axioms 2021, 10, 144 4 of 6 r r−1 (r) (r,0) Xn+r−1 + Xn+r−2 + ··· + Xn = a ∑ Fn−r+j + ∑ tn+j j=1 j=0 (r) (r,0) = aFn+1 + tn+r = Xn+r. (r,a) Therefore, (Xn)n and (tn )n satisfy the same r-order recurrence relation.