Fibonacci q-gaussian sequences

Some results on average of Fibonacci and Lucas sequences

In this paper, we discuss about some results on average of Fibonacci and Lucas sequences that may be found in the OEIS code A111035.

Fibonacci-like sequences for variants of the tower of Hanoi, and corresponding graphs and gray codes

We modify the rules of the classical Tower of Hanoi puzzle in a quite natural way to get the Fibonacci sequence involved in the optimal algorithm of resolution, and show some nice properties of such a variant. In particular, we deduce from this Tower of Hanoi-Fibonacci a Gray-like code on the set of binary words without the factor 11, which has some properties intersting for itself and from which an iterative algorithm for the Tower of Hanoi-Fibonacci is obtained. Such an algorithm involves the Fibonacci substitution. Eventually, we briefly extend the study to some natural generalizations.

Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes

We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

A $q$-analogue of the Biperiodic Fibonacci Sequence

The Fibonacci sequence has been generalized in many ways. One of them is defined by the relation $t_n=at_{n-1}+t_{n-2}$ if $n$ is even, $t_n=bt_{n-1}+t_{n-2}$ if $n$ is odd, with initial values $t_0=0$ and $t_1=1$, where $a$ and $b$ are positive integers. This sequence is called biperiodic Fibonacci sequence. In this paper, we introduce a $q$-analogue of this sequence. We prove several identities of $q$-analogues of the Fibonacci sequence. We give algebraic and combinatorial proofs.

广义斐波那契序列在网格中的一个新性质

A Novel Property of Generalized Fibonacci Sequence in Grids

本文研究了奇数阶网格中主对角线与副对角线上数字和的比值，揭示其仅依赖于网格阶数，提出一个简洁优雅的恒等式。

Optimal pebbling of complete binary trees and a meta-Fibonacci sequence

In this paper, we present a new method for determining the optimal pebbling number of a complete binary tree. This method reveals a curious connection between the optimal pebbling numbers of complete binary trees and the Conolly-Fox sequence, a type of meta-Fibonacci sequence.

Shadow sequences of integers, from Fibonacci to Markov and back

The following general idea looks crazy. What if another integer sequence follows each integer sequence like a shadow? I will demonstrate that this is indeed the case, perhaps not for every integer sequence, but for many of them.

Fibonacci自动序列的线性子序列复杂性

Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

研究Fibonacci自动序列的线性子序列的复杂性，分析自动机状态数及基本操作的复杂性，改进Bosma和Don的界。

Fibonacci-run graphs II: Degree sequences

Fibonacci cubes are induced subgraphs of hypercube graphs obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s. This class of graphs has been studied extensively and generalized in many different directions. Induced subgraphs of the hypercube on binary strings with restricted runlengths as vertices define Fibonacci-run graphs. These graphs have the same number of vertices as Fibonacci cubes, but fewer edges and different graph theoretical properties. Basic properties of Fibonacci-run graphs are presented in a companion paper, while in this paper we…

斐波那契数列增加一个斐波那契数后的扩展 Frobenius 问题

The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci number

本文研究由斐波那契数列平移一个斐波那契数所得序列的扩展 Frobenius 问题，并证明相应数值半群族满足 Wilf 猜想。

Linear Recurrent Subsequences of Meta-Fibonacci Sequences

In a recent paper, Frank Ruskey asked whether every linear recurrent sequence can occur in some solution of a meta-Fibonacci sequence. In this paper, we answer his question in the affirmative for recurrences with positive coefficients.

旋转记忆斐波那契数与周期平铺

Rotating-Memory Fibonacci Numbers and Periodic Tilings

本文提出旋转记忆斐波那契数，一种递推记忆周期性变化的数列，证明其刚性结构、闭式与生成函数，并通过周期平铺给出几何解释，为更高周期建模提出开放问题。

Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

$N$-factor complexity of the infinite Fibonacci sequence and digital sequences

In this paper, we introduce a variation of the factor complexity, called the $N$-factor complexity, which allows us to characterize the complexity of sequences on an infinite alphabet. We evaluate precisely the $N$-factor complexity for the infinite Fibonacci sequence $\mathbf{f}$ given by Zhang, Wen and Wu [Electron. J. Comb., 24 (2017)]. The $N$-factor complexity of a class of digit sequences, whose $n$th term is defined to be the number of occurrences of a given block in the base-$k$ representation of $n$, is also discussed.

Generalized Fibonacci recurrences and the lex-least De Bruijn sequence

The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographically-least de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and Lucas numbers, respectively.

Free Fibonacci Sequences

This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze these sequences for small n: 2, 3, 4, and 5. Surprisingly these behaviors are very different. We also talk about any n. Many statements about these sequences are difficult or impossible to prove, but they can be supported by probabilistic arguments, we have plenty of those in this paper. We also introduce ten new sequences. Most of the new sequences are also…

Fibonacci sequence related to a combinatorial problem on binary matrices

We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of integer sequences related to the discussed problem. The obtained sequences are new and they are not described in the On-Line Encyclopedia of Integer Sequences (OEIS). We show a relationship between some particular values of the parameters and the Fibonacci sequence.

一维、二维单词的Fibonacci序列：不动点因子的枚举与定位

Fibonacci Sequences of 1D, 2D Words: Enumerating and Locating the Factors of the Fixed Points

该研究针对二维单词因子枚举难题，探究了两种Fibonacci阵列序列不动点因子的枚举方法与一种定位方法，并分析了其因子复杂度与因子位置，助力理解二维单词结构。

Some enumerative properties of a class of Fibonacci-like cubes

A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in this paper. Several enumerative polynomials, e.g.\ rank generating function, cube polynomials and degree sequence polynomials are obtained. Some of these results relate to Fibonacci sequence and Padovan sequence.

Spot-Based Generations for Meta-Fibonacci Sequences

For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves "similarly" in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept - the spot-based generation sequence - to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences…

A family of meta-Fibonacci sequences defined by variable order recursions

We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and lower bounds which depend only on r(n).

斐波那契字的广义自然密度$\text{DF}(\frak{F}_n)$

Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word

本文探讨斐波那契序列的广义化，证明随机斐波那契序列项绝对值的n次方根收敛值、建立斐波那契字相关对称恒等式及无穷级数恒等式，提出斐波那契字密度新概念，揭示数论与组合学关联。

An upper bound on asymptotic repetitive threshold of balanced sequences via colouring of the Fibonacci sequence

We colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetitive threshold of $d$-ary balanced sequences. The bound is attained for $d=2, 4$ and $8$ and we conjecture that it happens for infinitely many even $d$'s. Our bound reveals an essential difference in behavior of the repetitive threshold and the asymptotic repetitive threshold of balanced sequences. The repetitive threshold of $d$-ary balanced sequences is known to be at least $1+\frac{1}{d-2}$ for each $d \geq 3$. In contrast, our bound implies that the asymptotic repetitive th…

Partial Sums of the Fibonacci Sequence

Let $(F_n)_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n): = (\sum_{i=1}^n F_i)_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is the resulting sequence from applying $P$ to $(F_n)$ $k$ times. Second, we provide a combinatorial interpretation of the numbers in $P^k(F_n)$.

广义斐波那契序列的多项式恒等式

Polynomial Identities for Generalized Fibonacci Sequences

该研究针对斐波那契、卢卡斯等序列的多项式恒等式，补充了广义斐波那契序列的相关恒等式，通过吉拉尔-沃林公式、比内公式结合线性代数两种方法完成序列分类，填补了数学文献中该类恒等式通用表述的空白。

Conway's subprime Fibonacci sequences

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.

Fibonacci colored compositions and applications

We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and quaternary words, spanning trees in the ladder graph, unimodal sequences covering an initial interval, and ordered-consecutive partition sequences. Our approach relies on the basic idea of representing the colored compositions as tilings of an $n$-board whose tiles are connected, and sometimes decorated, according to a suitable combinatorial interpretation of the giv…

Asymptotics of Lagged Fibonacci Sequences

Consider "lagged" Fibonacci sequences $a(n) = a(n-1)+a(\lfloor n/k\rfloor)$ for $k > 1$. We show that $\lim_{n\to\infty} a(kn)/a(n)\cdot\ln n/n = k\ln k$ and we demonstrate the slow numerical convergence to this limit and how to deal with this slow convergence. We also discuss the connection between two classical results of N.G. de Bruijn and K. Mahler on the asymptotics of $a(n)$.

Kac-Moody斐波那契序列

Kac--Moody Fibonacci sequences

本文研究了通过Kac-Moody代数根格生成无限整数序列的方法，并计算了部分序列的前二十项。

Fibonacci 型划分及其相关的分段定义置换

Fibonacci-like partitions and their associated piecewise-defined permutations

本文引入 Fibonacci 型划分，细化 Wythoff 划分，研究其上的分段置换，并应用于 Shapovalov 和 Venkatachala 的序列，推广得到猜想产生无限正交阵列的序列族。

Matrix Powers of Column-Justified Pascal Triangles and Fibonacci Sequences

If L, respectively R are matrices with entries binom{i-1,j-1}, respectively binom{i-1,n-j}, it is known that L^2 = I (mod 2), respectively R^3 = I (mod 2), where I is the identity matrix of dimension n > 1 (see P10735-May 1999 issue of the American Mathematical Monthly). We generalize it for any prime p, and give a beautiful connection to Fibonacci numbers.

Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers

In this paper, we consider nine OEIS sequences, the analysis of which allows us to find a connection between Motzkin numbers and Fibonacci numbers. In each Motzkin number, we distinguish an even component and an odd component, the difference of these two components is called the shadow of the Motzkin number. Reverse of the Motzkin shadows give us a sequence from the family of Fibonacci numbers.

Recurrence algorithms of waiting time for the success run of length $k$ in relation to generalized Fibonacci sequences

Let $V(k)$ denote the waiting time, the number of trials needed to get a consecutive $k$ ones. We propose recurrence algorithms for the probability distribution function (pdf) and the probability generating function (pgf) of $V(k)$ in sequences of independent and Markov dependent Bernoulli trials using generalized Fibonacci sequences of order $k$. Maximum likelihood estimation (MLE) methods for the probability distributions are presented in both cases with simulation examples.

Morphic words, Beatty sequences and integer images of the Fibonacci language

Morphic words are letter-to-letter images of fixed points $x$ of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of $x$. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called $AA$ sequences, where $α$ is a quadratic irrational, $A$ is the Beatty sequence defined by $A(n)=\lfloor αn\rfloor$, and $AA$ is…

Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond

Let $α= (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i α\rfloor$, $b_i = \lfloor i α^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form $a_i + a_j$, $b_i + b_j$, $a_i + b_j$, and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented.…

A Note on the Fibonacci Sequence and Schreier-type Sets

A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}_{n, p, q}|)_{n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$), where $$\mathcal{K}_{n, p, q} \ = \ \{A\subset \{1, \ldots, n\}\,:\, \mbox{either }A = \emptyset \mbox{ or } (\max A-\max_2 A = p\mbox{ and }\min A\ge |A|\ge q)\}$$ and $\max_2 A$ is the second largest integer in $A$, given that $|A|\ge 2$. When $p = 1$ and $q=2$, we have that $(|\mathcal{K}_{n, 1, 2}|)_{n=1}^\infty$ is the Fibonacci sequence. As a corollary…

Asymptotics of the Euler transform of Fibonacci numbers

The generating function for the sequence A166861 in the OEIS (Euler transform of Fibonacci numbers) is Product_{k>0} 1/(1-x^k)^F(k), where F(k) are the Fibonacci numbers. This paper analyzes the more general generating function U(x) = Product_{k>0} 1/(1-x^k)^F(k+z), where z is a nonnegative integer, which provides asymptotics for the sequences A166861 (z=0), A200544 (z=1) and A260787 (z=2) in the OEIS.

Fibonacci connection between Huffman codes and Wythoff array

Fibonacci connection between non-decreasing sequences of positive integers producing maximum height Huffman trees and the Wythoff array has been proved.

A Fibonacci sequence for linear structures with two types of components

We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to count, up to isomorphism, the number of these voting systems for an arbitrary number of voters. We obtain a closed formula for the number of these systems, this formula follows a Fibonacci sequence with a smooth polynomial variation on the number of voters.

关于双侧广义斐波那契数列的线性空间

On the linear space of the two-sided generalized Fibonacci sequences

本文研究了满足 $k$ 阶递推方程的双侧广义斐波那契数列的线性空间，并给出了分别基于广义二项式系数和广义多项式系数的两类显式公式。

斐波那契数及相关序列上分拆的渐近分析

Asymptotics for partitions over the Fibonacci numbers and related sequences

本文结合经典分拆理论与 Fibonacci Dirichlet 级数的新结果，首次给出整数表示为非互异斐波那契数之和的方式数 $p_F(n)$ 的渐近公式，并将类似结果推广到一般线性递推序列。

The number of valid factorizations of Fibonacci prefixes

We establish several recurrence relations and an explicit formula for V(n), the number of factorizations of the length-n prefix of the Fibonacci word into a (not necessarily strictly) decreasing sequence of standard Fibonacci words. In particular, we show that the sequence V(n) is the shuffle of the ceilings of two linear functions of n.

Areas of Triangles and other Polygons with Vertices from Various Sequences

Motivated by Elementary Problem B-1172 in the Fibonacci Quarterly (vol. 53, no. 3, pg. 273), formulas for the areas of triangles and other polygons having vertices with coordinates taken from various sequences of integers are obtained. The sequences discussed are Polygonal number sequences as well as Fibonacci, Lucas, Jacobsthal, Jacobsthal-Lucas, Pell, Pell-Lucas, and Generalized Fibonacci sequences. The polygons have vertices with the form $(p_n,p_{n+k}),\,\, (p_{n+2k},p_{n+3k}),\,\,\dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.

Isoperimetric Sequences for Infinite Complete Binary Trees, Meta-Fibonacci Sequences and Signed Almost Binary Partitions

In this paper we demonstrate connections between three seemingly unrelated concepts. (1) The discrete isoperimetric problem in the infinite binary tree with all the leaves at the same level, $ {\mathcal T}_{\infty}$: The $n$-th edge isoperimetric number $δ(n)$ is defined to be $\min_{|S|=n, S \subset V({\mathcal T}_{\infty})} |(S,\bar{S})|$, where $(S,\bar{S})$ is the set of edges in the cut defined by $S$. (2) Signed almost binary partitions: This is the special case of the coin-changing problem where the coins are drawn from the set ${\pm (2^d - 1): $d$ is a positive integer}$. The quantity …

斐波那契-富比尼数与卢卡斯-富比尼数

The Fibonacci-Fubini and Lucas-Fubini numbers

本文通过将有序贝尔数中的划分块按斐波那契方式排列，引入斐波那契-富比尼数，并研究其生成函数、显式公式、组合解释及算术三角形，同时揭示其与斯特林数的联系。

Splitting fields and periods of Fibonacci sequences modulo primes

What is the period of the Fibonacci sequence modulo a prime? The purpose of our brief expository paper is to illustrate an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra (rather than the case-by-case analysis found in many papers on the subject, or techniques from graduate number theory). Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in undergraduate course in abstract algebra or …

关于一个 Fibonacci–Lucas 恒等式的若干注记

Some notes on a Fibonacci-Lucas identity

本文围绕 Edgar–Bhatnagar 的 Fibonacci–Lucas 多项式恒等式，为广义 Lucas 序列提出新的多项式恒等式、特例及 Chebyshev 多项式推广。

两类广义Fibonacci子序列的渐近估计

The asymptotic estimation for two classes of generalized Fibonacci sub-sequences

本文利用初等方法，对广义Fibonacci序列的两类无穷级数倒数的渐近估计进行了研究，推广了Yuan等人2025年的结果。

通用斐波那契序列与UFS-群oid

Universal Fibonacci sequences and UFS-groupoids

本文研究了允许通用斐波那契序列的UFS-群oid，证明了其性质并给出了有限和无限情况下的分类与构造。

关于斐波那契密度上无限词 $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ 的改进遍历理论

Improvement Ergodic Theory For The Infinite Word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ on Fibonacci Density

本文研究斐波那契密度上无限词的遍历理论，分析其子词多样性与广义模式分布，并揭示替换规则、代数数论与组合复杂性之间的深层联系。

台球分拆、斐波那契数列、SIP类与箭图

Billiard Partitions, Fibonacci Sequences, SIP Classes, and Quivers

本文从椭球台球周期轨道中的台球分拆出发，引入B型可分离整数分拆类，研究其基本分拆数与斐波那契数列的关系，并建立其生成级数与对称箭图生成级数及Schröder路径计数的联系。

Zeckendorf expansion, Dirichlet series and infinite series involving the infinite Fibonacci word

Let $β=\frac{1+\sqrt{5}}{2}$, $(a_n)_{n \in \mathbb{N}^+}$ be a non-uniform morphic sequence involving the infinite Fibonacci word and $(δ(n))_{n \in \mathbb{N}^+}$ be a positive sequence such that for all positive integers $n$, $δ(n)=\frac{1}{\sqrt{5}}\sum_{j \geq 0}ε_jβ^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n=\sum_{j \geq 0}ε_jF_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of $\sum_{n\geq 1}\frac{a_n}{(δ(n))^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fib…