Related papers: On monotonicity of some combinatorial sequences

Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci…

For a fixed irrational number $\alpha$ and $n\in \mathbb{N}$, we look at the shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where $f(i) = \alpha i \mod 1$. Our primary result is that the boundary of the Schensted…

A new short clear proof of the asymptotics for the number $c_n$ of real roots of the Bernoulli polynomials $B_n(x)$, as well as for the maximal root $y_n$: $$y_n=\frac{n}{2\pi e}+\frac{\ln(n)}{4\pi e}+O(1)\quad\text{and}\quad…

Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m>=3, there are systems of the form (f,g) having exactly 2m-1 roots in the positive quadrant. Even examples with m=4 having 7…

In this paper, we study natural numbers $m$ with ${u_{z \pm 1} \equiv -1 \ (m)}$ for a $z \in \mathbb{N}$, where $u_n$ is the $n$th Fibonacci number. Furthermore, we want to show for $m \in \mathbb{N}\setminus\{0,1\}$: $[\forall p \…

I propose and investigate the use of continuous functional equations for the study of meta-Fibonacci integer sequences. This exploratory study includes three sequences with quite different behavior: Conway's famous sequence $A(n)=…

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the…

The function f:X -> Y is called k-monotonically increasing if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is k-monotonically…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves…

We consider random polynomials $p_n(x)=\xi_0+\xi_1+\dots+\xi_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment (for some $\epsilon>0$), also known as…

In this article, we study the decay rates of the correlation functions for a hyperbolic system $T: M \to M$ with singularities that preserves a unique mixing SRB measure $\mu$. We prove that, under some general assumptions, the correlations…

Let A be a finite set of integers. We prove that if |A| is at least 2 and |A+A| is 3|A|-3, then one of the following is true: 1. A is a bi-arithmetic progression; 2. A+A contains an arithmetic progression of length 2|A|-1; 3. |A| is 6 and A…

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-$n$…

We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the…

Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$…

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…

Harmonic numbers are important in a lot of branches of number theory. By means of the derivative operator, the integral operator, and several summation and transformation formulas for hypergeometric series, we prove four series containing…

An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one…

We give a negative answer to a question by Paul Erd\H{o}s and Ronald Graham on whether the series \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots(n+f(n))} \] has an irrational sum whenever $(f(n))_{n=1}^{\infty}$ is a sequence of positive…