Related papers: A product rule for triangular numbers

We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama gave a precise criterion for when the tensor product of an $n$-representation finite algebra…

Using T(m,n;k) to denote the number of ways to make a selection of k squares from an (m x n) rectangular grid with no two squares in the selection adjacent, we give a formula for T(2,n;k), prove some identities satisfied by these numbers,…

We discuss the problem to count, or, more modestly, to estimate the number f(m,n) of unimodular triangulations of the planar grid of size $m\times n$. Among other tools, we employ recursions that allow one to compute the (huge) number of…

A quick way to compute generating functions related to Pell-Padovan tetranacci numbers and classical sequences of recursions of order two is provided. Eight special instances can be computed at once.

In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an…

When ${\bf t} := \langle t_1,t_2,\ldots,t_k\rangle$ is a sequence of transpositions on the finite set $n:=\{0,1,\ldots,n-1\}$, then $\bigcirc{\bf t}:= t_1\circ t_2\circ\cdots\circ t_k$ denotes the compositional product of the sequence. Our…

Let $M_n(R)$ be the algebra of all $n\times n$ matrices over a unital commutative ring $R$ with 6 invertible. We say that $A\in M_n(R)$ is a Jordan product determined point if for every $R$-module $X$ and every symmetric $R$-bilinear map…

This paper presents iterative methods for solving tensor equations involving the T-product. The proposed approaches apply tensor computations without matrix construction. For each initial tensor, these algorithms solve related problems in a…

This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(\alpha)$, where $S(\alpha)$ is the set of all integers of the form $\lfloor…

This work provides a complete characterization of congruent numbers in terms of Pythagorean triples. Specifically, we show that every congruent number can be written as $$\frac{nm\left(m-n\right)\left(m+n\right)}{\sigma^2}$$ were as…

Infinite products associated with the $\pm 1$ Thue-Morse sequence whose value is rational or algebraic irrational have been studied by several authors. In this short note we prove three new infinite product identities involving ${\pi}$,…

Let $\sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$…

In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of $n$-cycles whose product has $k$ cycles and has $m$ given elements contained in distinct…

Let $G$ be a finite group. A sequence over $G$ means a finite sequence of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product…

We determine couples of singular moduli which have rational products

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

In this paper, we prove that for any $a,M\in \mathbb N$ with $(a,M)=1$, there are infinitely many Carmichael numbers $m$ such that $m\equiv a$ mod $M$

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack…

It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product $a\circ b = \sqrt{a}b\sqrt{a}$ on an operator…