Christian Ballot
We survey and prove properties a family of recurrences bears in relation to integer representations, compositions, the Pascal triangle, sums of digits, Nim games and Beatty sequences.
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In this paper,we find all generalized Fibonacci numbers which are Narayana's cows numbers. In our proofs, we use both Baker's theory of nonzero linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction method.
Narayana numbers that are products of two Fibonacci numbers
本文求解丢番图方程 $N_m=F_nF_k$，确定所有可表示为两个斐波那契数乘积的非零 Narayana 数。
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given.
We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the $3$-adic valuation of ${a_n}$ and use this to determine all Narayana numbers that are factorials.
A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M. Lasalle is then obtained from a classical convolution identity. Some arithmetical properties of the generalized coefficients are also established.
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several generalizations of this sequence and also several interesting identities. In this paper, we investigate a homogeneous recurrence relation that, in a way, extends the linear recurrence of the Fibonacci sequence for two variables, called {\it double-recurrence Fibonacci numbers}, given by ${F(m,n) = F(m-1, n-1)+F (m-2, n-2)}$, for $n,m\geq 2$, where $F (m, 0) = F_m$, $F (m, 1) = F_{m+1}$, $F (0, n) = F_n$ and $F (1, n) = F_{n+1}$. We exhibit a…
Let $ (N_n)_{n\ge 0}$ be the Narayana's cow sequence defined by a third-order recurrence relation $ N_0=0,\ N_1= N_2=1 $, and $ N_{n+3}=N_{n+2}+N_n $ for all $ n\ge 0 $. In this paper, we determine all Narayana numbers that are concatenations of two repdigits. The proof of our main theorem uses lower bounds for linear forms in logarithms and a version of the Baker-Davenport reduction method in Diophantine approximation.
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph associated to the recurrence relation.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
We derive the double recurrence $e_n = \frac{1}{2}(a_{n-1}+5b_{n-1}); f_{n} = \frac{1}{2}(a_{n-1}+b_{n-1})$ with $e_0=2;f_0=0$ for the Fibonacci numbers, leading to an extremely simple and fast implementation. Though the recurrence is probably not new, we have not been able to find a reference for it.
In this paper, we define the incomplete h(x)-Fibonacci and h(x)-Lucas polynomials, we study recurrence relations and some properties of these polynomials
In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type.
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations.
We present numerous interesting, mostly new, results involving the $n$-step Fibonacci numbers and $n$-step Lucas numbers and a generalization. Properties considered include recurrence relations, summation identities, including binomial and double binomial summation identities, partial sums and ordinary generating functions. Explicit examples are given for small $n$ values.
In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where $p$ is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, $m$ and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.
We give a simple condition for a linear recurrence (mod 2^w) of degree r to have the maximal possible period 2^(w-1).(2^r-1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for 3-term linear recurrences defined by trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.
We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integer versions of these sequences, where the underlying prime congruence structures of the rational sequence denominators enables the integer sequence to be recovered. From either the rational or the integer sequences we construct sequences of vectors in $\mathbb{Q}^m$, which converge to irrational algebraic points in $\mathbb{R}^m$. The rational sequence terms can be expressed as simple recurren…
In this paper, we consider the new family of recurrence sequences of $(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of $k$-generalized Fibonacci numbers and generalized $k$-order Pell numbers. We shall obtain a Binet-style formula and study the asymptotic behavior of dominant root of characteristic equation. Moreover, we shall prove some auxiliary results about these sequences. In particular, we characterize the first $(q,k)$-generalized Fibonacci numbers in terms of binary sequences.
Representations with k-generalized Fibonacci numbers
该研究探讨整数的k阶广义斐波那契数表示，给出系数为{-1,0,1}的零表示计数的递推关系，构造系数为{0,1}的 tribonacci 表示的二叉树模型，揭示其与自相似测度的联系。
By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent elements; this is called a legal decomposition. Previous work examined the distribution of the number of summands and the spacings between them, in legal decompositions arising from the Fibonacci numbers and other linear recurrence relations with non-negative integral coefficients. Many of these results were restricted to the case where the first term in th…
Two Primes and Five Mixed Fibonacci-Lucas Summands
该研究用圆法证明，对u=1至4，每个足够大的正整数可表为两素数加u个斐波那契数与5−u个卢卡斯数之和，项可重复且递推指标为正。
Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the $(s,b)$-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the $(s,b)$-Generacci s…
Recurrence Relations for $S$-Legal Index Difference Sequences
本研究推广Fibonacci Quilt序列的构造，提出$S$-合法索引差（$S$-LID）序列，证明其增长规律及多类集合$S$对应的序列满足简单递推关系。
In 1989, Ming Luo \cite{L2} showed that the Fibonacci number $U_n$ is Triangular if and only if $n=\pm1,2,4,8,10$. For this, he established a Jacobi Symbol Criterion. Moreover, he observed that this problem is equivalent to finding all integer points on two elliptic curves. In this paper, we prove a Jacobi Symbol Criterion for more general families of binary recurrences. In addition, applying the criterion and elementary methods, we determine all integer points on the elliptic curves $y^2=5x^2(x+3)^2+4(-1)^n$.
A common generalization of Dickson polynomials, Fibonacci polynomials, and Lucas polynomials and applications
本文定义了满足线性递推关系的更一般的多变量多项式族，给出了显式公式和行列式表达式，并将其应用于2阶递推多项式，揭示了Fibonacci、Lucas和Dickson多项式等的关系与恒等式，是对多项研究的统一推广。
In this study, the new algebraic properties related to bivariate Fibonacci polynomials has been given. We present the partial derivatives of these polynomials in the form of convolution of bivariate Fibonacci polynomials. Also, we define a new recurrence relation for the r-th partial derivative sequence of bivariate Fibonacci polynomials.
Non-standard Zeckendorf decompositions; or, Tribonacci within Fibonacci
研究用前k个Fibonacci数表示n的方式数B(n;k)，展示B(0;k)满足类似Tribonacci的递推关系，B(n;k)满足修改后的递推关系。
We prove recurrence relations and modulo periodic properties of multiple derivatives of Fibonacci polynomials. We apply the obtained results to present the dynamic structures of Fibonacci polynomials over the ring of 2-adic integers by investigating minimal decompositions which consist of minimal subsystems and attracting basins.
In this paper, we consider a generalization of Horadam sequence fwng which is defined by the recurrence relation wn = x(n)wn-1+ cwn-2; where x(n) = a if n is even, x(n) = b if n is odd with arbitrary initial conditions w0;w1 and nonzero real numbers a, b and c. As a special case, by taking initial conditions 0, 1 and 2, b we define the sequences {un} and {vn}, respectively. The main purpose of this study is to derive some basic properties of the sequences {un}, {vn} and {wn} by using matrix approach.
Zeckendorf proved that every positive integer can be expressed as the sum of non-consecutive Fibonacci numbers. This theorem inspired a beautiful game, the Zeckendorf Game. Two players begin with $n \ 1$'s and take turns applying rules inspired by the Fibonacci recurrence, $F_{n+1} = F_n + F_{n-1}$, until a decomposition without consecutive terms is reached; whoever makes the last move wins. We look at a game resulting from a generalization of the Fibonacci numbers, the Fibonacci Quilt sequence. These arise from the two-dimensional geometric property of tiling the plane through the Fibonacci s…
Polynomial values with integer coefficients of the generating functions for Fibonacci polynomials
本文将Fibonacci序列生成函数取整数值的判别结果推广到满足带系数递推关系的Fibonacci多项式，并在$\sqrt d$代入时给出同形式结论。
Meta-automatic Sequences
本文提出元自动序列的概念，结合元斐波那契递归与自动序列的基于数字的递归，研究两个二进制元自动序列M1和M2的DFAO评估、4-一致态射及其因子复杂度。
A method for obtaining Fibonacci identities
本文讨论了卢卡斯序列的恒等式，提出了一种获取递推序列恒等式的方法，并发现了一种插值类型的二阶线性递推关系恒等式。
The generalized Fibonacci sequences are sequences $\{f_n\}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t \in \mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan, Chen, Moll, and Sagan considered some arithmetic properites of the generalized Fibonacci sequence. Specifically, they considered the behavior of analogues of the $p$-adic valuation and the Riemann zeta function. In this paper, we resolve some conjectures which they raised relating to these topics. We also consider the rank modulo $n$ in more d…
Fixed points of K-Fibonacci sequences
本文研究了 $K$-Fibonacci 序列的不动点，根据 $K^2+4$ 的因式分解确定了所有不动点，并证明迭代该过程后所有大于 3 的模数最终都会终止于不动点。
A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the coefficients of the Fibonacci numbers in the decomposition to be zero or $\pm 1$, the resulting expression is known as the far-difference representation. Alpert proved that a far-difference representation exists and is unique under certain restraints that generalize non-consecutiveness, specifically that two adjacent summands of the same sign must be at lea…
Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even indexed Fibonacci numbers. We extend these further and prove that similar results hold when only using indices in a given arithmetic progression. As part of our proofs, we generate a range of new recurrences for the Fibonacci numbers that are of interest in their own right.
In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated.
Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from $1, 2, 3, 5,\ldots$. This has been generalized by many authors, in particular to constant coefficient fixed depth linear recurrences with positive (or in some cases non-negative) coefficients. In this work we extend this result to a recurrence with non-constant coefficients, $a_{n+1} = n a_{n} + a_{n-1}$. The decomposition law becomes every $m$ has a unique decomposition as $\sum s_i a_i$ with $s_i \le i$, where if $s_i = i$ then $s_{i-1} = 0$. Similar to…
Given a recurrent sequence ${\bf U}:=\{U_n\}_{n\ge 0}$ we consider the problem of counting ${\mathcal M}_U(x)$, the number of integers $n\le x$ such that $U_n=u^2+nv^2$ for some integers $u,v$. We will show that ${\mathcal M}_U(x)\ll x(\log x)^{-0.05}$ for a large class of ternary sequences. Our method uses many ingredients from the proof of Alba González and the second author that ${\mathcal M}_F(x)\ll x(\log x)^{-0.06}$, with $\bf F$ the Fibonacci sequence.
Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers
该研究提出k-广义斐波那契数Fₖ(n)的新表达式，利用Fuss-Catalan数给出其特征多项式根的显式形式，量化渐近近似精度，提供任意初始条件下的递推基，扩展前人结果并发现多变量Dickson多项式关联及伴随矩阵新恒等式。
It's the age-old recurrence with a twist: sum the last two terms and if the result is composite, divide by its smallest prime divisor to get the next term (e.g., 0, 1, 1, 2, 3, 5, 4, 3, 7, ...). These sequences exhibit pseudo-random behaviour and generally terminate in a handful of cycles, properties reminiscent of 3x+1 and related sequences. We examine the elementary properties of these 'subprime' Fibonacci sequences.
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials that divide their compositions with every positive integer power. We completely characterize such polynomials in terms of their factorizations into cyclotomic polynomials using labeled Hasse diagrams, and construct new integer divisibility sequences based on them. We also show that, unlike the Fibonacci numbers, these non-classical sequences do not have the p…
On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- II
本文研究了任意起始项与偶数步长下的广义Fibonacci子列集合，确定了其Frobenius数和亏格的计算问题。
The only recurrence sequence satisfying the Fibonacci recurrence and realizable as the number of periodic points of a map is (a multiple of) the Lucas sequence.
Modular Periodicity of Random Initialized Recurrences
研究所有可能初始值下随机初始化递推的周期结构，发现斐波那契递推与其奇偶变换的镜像对称性，揭示素数到素数幂模的分形自相似性，并定义由卢卡斯比值支配的极小分布。
We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci sequence, Pell sequence, Jacobsthal sequence, and the Balancing sequence of numbers. The paper also provides several approaches in solving the linear recurrence relation under consideration. We end the paper by giving out an open problem.
On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- I
本文研究广义斐波那契子序列构成的集合的 Frobenius 数与亏格，针对任意起始项及奇数或等于2的间隔参数，给出计算方法。
There is a growing literature on sums of reciprocals of polynomial functions of recurrence relations with constant coefficients and fixed depth, such as Fibonacci and Tribonacci numbers, products of such numbers, and balancing numbers (numbers $n$ such that the sum of the integers less than $n$ equals the sum of the $r$ integers immediately after, for some $r$ which is called the balancer of $n$; If $n$ is included in the summation, we have the cobalancing numbers, and $r$ is called the cobalancer of $n$). We generalize previous work to reciprocal sums of depth two recurrence sequences with ar…
In this paper, we present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P,Q,R_0$, and $R_1$ be fixed integers and let $\boldsymbol{R}=\left(R_n\right)_{n}$ be the recurrence sequence defined by $R_{n+2}=PR_{n+1}-QR_{n}$ $(\forall n\geq 0)$. Under some conditions on the parameters, we determine a rational nontrivial divisor for $L_{k,n}:=\mathrm{lcm}\left(R_k,R_{k+1},\dots,R_n\right)$, for all positive integers $n$ and $k$, such that $n\geq k$. As consequences, we derive nontrivial effective lower bounds for $L_{k,n}$ and we estab…
For a positive integer $n$, we study the number of steps to reach $n$ by a {\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the recurrence of $a_{k+2}=a_{k+1}+a_k$. The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number $s(n)$ of steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property that $a_{s+1}$ is the integer closest to $ϕn$ where $ϕ=(1+\sqrt{5})/2$. We prove that this is true for only a positive fraction of $n$. We gi…
Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result, though with a different notion of a legal decomposition, holds for many other sequences. We use these decompositions to construct a two-player game, which can be completely analyzed for linear recurrence relations of the form $G_n = \sum_{i=1}^{k} c G_{n-i}$ for a fixed positive integer $c$ ($c=k-1=1$ gives the Fibonaccis). Given a fixed integer $n$ and an initial decomposition of $n = n G_1$, the two players alternate by using moves related to the recurrenc…
Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence
本研究推导了一类三对角实对称矩阵特征多项式的三项递推关系与闭式解，揭示其系数与帕斯卡三角对角元的关联，给出特征值包含关系的充分条件，并提出关联斐波那契数列的开放问题。
In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0, u_1=1 and u_{i+1}=au_i+u_{i-1} for i=1,2,3,..., then all the sums u_m+au_n (m,n=1,2,3,...) are distinct. One of our conjectures states that any integer n>4 can be written as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number.
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