Related papers: Factorizations of the Fibonacci Infinite Word
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
We introduce the notion of $\alpha$-numbers and formal intercept of sturmian words, and derive from this study general factorisations formula for sturmian words. Sturmian words are defined as infinite words with lowest unbound complexity,…
In part 1 of this paper some linear weighted generalized Fibonacci number summation identities were derived using the fact that the Fibonacci number is the residue of a rational function. In this part, using the same method, some quadratic…
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…
In this article, we will discover some new generalized identity regarding continued fractions. We will connect the results to Fibonacci numbers and Lucas numbers. For all the proof, we will use induction.
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a…
In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism…
We make an asymptotic analysis via singularity analysis of generating functions of a number sequence that involves the Fibonacci numbers and generalizes the binomial coefficients.
We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical…
We establish correspondances between factorisations of finite abelian groups (direct factors, unitary factors, non isomorphic subgroup classes) and factorisations of integer matrices. We then study counting functions associated to these…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
Fibonacci word fractals are a class of fractals that have been studied recently, though the word they are generated from is more widely studied in combinatorics. The Fibonacci word can be used to draw a curve which possesses…
We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization $x= U_0 U_1U_2 \cdots $ where each $U_i$ is a non-empty prefix of $x.$ With each…
For any given finite abelian group, we give factorizations of the group determinant in the group algebra of any subgroup. The factorizations are an extension of Dedekind's theorem. The extension leads to a generalization of Dedekind's…
Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.
The aim of this note is to show that Poincar\'e inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincar\'e inequalities are considered, too. The proof is short and does not involve covering…
We consider a sequence of sums of powers of the the roots of the cubic equation characterizing the Tribonacci sequences and derive its relationship with a particular Tribonacci sequence. Then we make a conjecture on the possible…
This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words,…
The sequence of partial sums of Fibonacci numbers, beginning with $2$, $4$, $7$, $12$, $20$, $33,\dots$, has several combinatorial interpretations (OEIS A000071). For instance, the $n$-th term in this sequence is the number of length-$n$…
This article presents a combinatorial result on indexed languages which was inspired by an attempt to understand the structure of groups with indexed language word problem. We show that a sufficiently long word in an indexed language can be…