Related papers: A Generalized Digit Map: Periodicity, Prouhet-Tarr…

We give a few remarks on the periodic sequence $a_n=\binom{n}{x}~(mod~m)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being…

Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \emph{matrix multiplicity} $r(n)$, the…

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that…

A graph is Hamiltonian if it contains a cycle which visits every vertex of the graph exactly once. In this paper, we consider the problem of Hamiltonicity of a graph $G_n$, which will be called the prime difference graph of order $n$, with…

In this paper we introduce a class of sequences connected with the $m$--ary partition function and investigate their congruence properties. In particular, we get facts about the sequences of $m$--ary partitions $(b_{m}(n))_{m\in\mathbb{N}}$…

Given an integer base $b>1$, a set of integers is represented in base $b$ by a language over $\{0,1,...,b-1\}$. The set is said to be $b$-recognisable if its representation is a regular language. It is known that eventually periodic sets…

Let $T(G;X,Y)$ be the Tutte polynomial for graphs. We study the sequence $t_{a,b}(n) = T(K_n;a,b)$ where $a,b$ are non-negative integers, and show that for every $\mu \in \N$ the sequence $t_{a,b}(n)$ is ultimately periodic modulo $\mu$…

Let $f_{n}=\sum_{i=0}^n \binom{n}{i}\binom{2n-2i}{n-i}$, $g_{n}= \sum_{i=1}^n \binom{n}{i}\binom{2n-2i}{n-i}$. Let $\{a_k\}_{k=1}$ be the set of all positive integers n, in increasing order, for which $\binom{2n}{n}$ is not divisible by 5,…

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences $x_{n+1}=F(x_n)$ generated by such maps display…

We study divide-and-conquer recurrences of the form \begin{equation*} f(n) = \alpha f(\lfloor \tfrac n2\rfloor) + \beta f(\lceil \tfrac n2\rceil) + g(n) \qquad(n\ge2), \end{equation*} with $g(n)$ and $f(1)$ given, where $\alpha,\beta\ge0$…

The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open…

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the…

Let $B_{u,v}(n)$ denote the number of $(u,v)$-regular bipartitions of $n$. In this article, we prove that $B_{p,m}(n)$ is always almost divisible by $p,$ where $p\geq 5$ is a prime number and $m=p_1^{\alpha_1} p_2^{\alpha_2}\cdots…

We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary…

Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with…

Arithmetical congruence monoids, which arise in non-unique factorization theory, are multiplicative monoids $M_{a,b}$ consisting of all positive integers $n$ satsfying $n \equiv a \bmod b$. In this paper, we examine the asymptotic behavior…

The long run behaviour of linear dynamical systems is often studied by looking at eventual properties of matrices and recurrences that underlie the system. A basic problem that lies at the core of many questions in this setting is the…

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…

Let $t_n = (-1)^{s_2(n)}$, where $s_2(n)$ is the sum of binary digits function. The sequence $(t_n)_{n\in \mathbb N}$ is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence $(h_n)_{n\in \mathbb…

Let $x_1,...,x_n$ be a list of real numbers, let $s :=\sum_{i=1}^{n}x_i$ and let $h:\mathbb{N} \rightarrow \mathbb{R}$ be a function. We gave a necessary and sufficient condition for $s>h(n)$ (respectively, $s<h(n)$). Let $G=(V,E)$ be a…