Related papers: Algorithms for complementary sequences

We show that for integers $k\geq 4$ and $s\geq k^2+(3k-1)/4$, we have an asymptotic formula for the number of solutions, in positive integers $x_i$, to the inequality $\left|(x_1-\theta_1)^k+\dotsc+(x_s-\theta_s)^k-\tau\right|<\eta$, where…

We derive some combinatorial formulas related to the diagonal Ramsey numbers $R(k)$. Each formula is a statement of the form "$F(n,k) = 0$ if and only if $n \ge R(k)$," where $F(n,k)$ is a combinatorial expression which depends on $n$ and…

We count the number of alignments of $N \ge 1$ sequences when match-up types are from a specified set $S\subseteq \mathbb{N}^N$. Equivalently, we count the number of nonnegative integer matrices whose rows sum to a given fixed vector and…

Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of $k$-generalized Fibonacci sequences…

We investigate the existence of closed polylines (also known as closed polygonal chains or self-crossing polygons) that intersect each of their edges the same number of times. The most general question in this corner of combinatorial…

In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. A non standard numerical method is developed to approximate the solutions in higher dimensional case. The idea…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only…

The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive…

In this paper we study the multiplicity of positive solutions for nonlinear elliptic equations on $\R^N$. The number of solutions is greater or equal than the number of disjoint intervals on which the nonlinear term is negative.…

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many summands are needed. Using a continued fraction approach,…

If a sequence indexed by nonnegative integers satisfies a linear recurrence without constant terms, one can extend the indices of the sequence to negative integers using the recurrence. Recently, Cigler and Krattenthaler showed that the…

A \Def{composition} of a positive integer $n$ is a $k$-tuple $(\l_1, \l_2, \dots, \l_k) \in \Z_{> 0}^k$ such that $n = \l_1 + \l_2 + \dots + \l_k$. Our goal is to enumerate those compositions whose parts $\l_1, \l_2, \dots, \l_k$ avoid a…

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd…

In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all…

This paper describes a new link between combinatorial number theory and geometry. The main result states that A is a finite set of relatively prime positive integers if and only if A = (K-K) \cap N, where K is a compact set of real numbers…

Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli…

In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the…