Related papers: Some additive relations in the Pascal triangle

This article demonstrates, using numerous examples of varying complexity, how one can visually prove summation formulas involving binomial coefficients by exclusively using the recurrence relation for binomial coefficients and its…

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…

The following work shows new connections between the constants $\pi$ and $e$ with Pascal's triangle and the Lucas triangle, established via Fibonacci polynomials and similar means. Furthermore, relations between the two famous constants and…

We give a generalization of the Pascal triangle called the quasi s-Pascal triangle where the sum of the elements crossing the diagonal rays produce the s-bonacci sequence. For this, consider a lattice path in the plane whose step set is {L…

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…

Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral…

We develop a random model for relation algebras. We prove some preliminary results and pose questions that lay out a new direction of research.

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find…

In this expository paper we collect some simple facts about analogues of Pascals triangle where the entries count subsets of the integers with an even or odd sum and show that they are related to Rogers-Szego polynomials. In particular we…

In this survey paper we discuss some recent results and related open questions in additive combinatorics, in particular, questions about sumsets in finite abelian groups.

A cohomology theory for "odd polygon" relations -- algebraic imitations of Pachner moves in dimensions 3, 5, ... -- is constructed. Manifold invariants based on polygon relations and nontrivial polygon cocycles are proposed. Example…

The alternating ternary sum in an associative algebra, $abc - acb - bac + bca + cab - cba$, gives rise to the partially alternating ternary sum in an associative dialgebra with products $\dashv$ and $\vdash$ by making the argument $a$ the…

Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is said to be an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. This…

We compute the sum and the alternating sum of the reciprocals of triangular numbers using two standard methods from calculus: a telescoping series approach and a power series approach. We then extend these results to generalized…

Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all…

Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…

We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this…