Related papers: Sums of Consecutive Integers

We study the automorphism group of the field of surreal numbers. Our main structure theorem presents a decomposition of this group into a product of five significant factors. Using the representation of surreal numbers as generalized power…

We determine the average number of distinct subsequences in a random binary string, and derive an estimate for the average number of distinct subsequences of a particular length.

We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

We study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations. When the iterates of the transformations are integer polynomials we prove that any such correlation sequence is the…

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple…

A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular…

We encode/decode Prolog terms as unique natural numbers. Our encodings have the following properties: a) are bijective b) natural numbers always decode to syntactically valid terms c) they work in low polynomial time in the bitsize of the…

Each natural number can be associated with some tree graph. Namely, a natural number $n$ can be factorized as $$ n = p_1^{\alpha_1}\ldots p_k^{\alpha_k},$$ where $p_i$ are distinct prime numbers. Since $\alpha_i$ are naturals, they can be…

Graph compositions generalize both integer compositions and partitions of a finite set. We develop formulas, generating functions and recurrence relations for composition counting functions for several families of graphs.

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

We study decompositions of NVALUE, a global constraint that can be used to model a wide range of problems where values need to be counted. Whilst decomposition typically hinders propagation, we identify one decomposition that maintains a…

We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.

Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…

A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti's Theorem. In its original form it applies to infinite 0-1 valued exchangeable sequences. Later it was extended and…

Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.

To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…

We discuss a formal system of mathematics. We use it to construct the natural numbers.

Consider $k\ge 2$ distinct, linearly independent, homogeneous linear recurrences of order $k$ satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree $k$, and there is a very broad…

We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…

We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.