A parametrization of equilateral triangles having integer coordinates

We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also show a similar characterization for the sides of a regular tetrahedron in $\Z^3$: such a tetrahedron exists if and only if the sides are of the form $k\sqrt{2}$, for some $k\in\N$. The classification of all the equilateral triangles in $\Z^3$ contained in a given plane is studied and the beginning analysis is presented. A more general parametrization is proven under a special assumption. Some related questions are stated in the end.Comment: 3 fugures, 17 pages, submitted to Integer

Half domination arrangements in regular and semi-regular tessellation type graphs

We study the problem of half-domination sets of vertices in vertex transitive infinite graphs generated by regular or semi-regular tessellations of the plane. In some cases, the results obtained are sharp and in the rest, we show upper bounds for the average densities of vertices in half-domination sets.Comment: 14 pages, 12 figure

A variation on bisecting the binomial coefficients

In this paper, we present an algorithm which allows us to search for all the bisections for the binomial coefficients $\{\binom{n}{k} \}_{k=0,...,n}$ and include a table with the results for all $n\le 154$. Connections with previous work on this topic is included. We conjecture that the probability of having only trivial solutions is $5/6$. \end{abstract}Comment: 14 pages, four tables, two figure

STATISTICAL ANNALISYS OF THE CORRELATION BETWEEN GDP, PRODUCTIVITY AND BRUTE INVESTMENTS AT THE LEVEL OF OLTENIA REGION*

This article try to put in debate the problem of obtaining a balanced development of the regions from a country by using correlation models which include variables like: GDP, productivity, brute investments etc. The purpose is to find the most important variables that can strongly influence the evolution of GDP at the regional level and also can be use to do that in practice. The base of this debate is a statistical analysis realized on Oltenia region at general level and at the level of almost every activity sectorsGDP, productivity, brute investments, correlation, statistical analysis

Counting all equilateral triangles in {0,1,2,...,n}^3

We describe a procedure of counting all equilateral triangles in the three dimensional space whose coordinates are allowed only in the set $\{0,1,...,n\}$. This sequence is denoted here by ET(n) and it has the entry A102698 in "The On-Line Encyclopedia of Integer Sequences". The procedure is implemented in Maple and its main idea is based on the results in \cite{eji}. Using this we calculated the values ET(n) for n=1..55 which are included here. Some facts and conjectures about this sequence are stated. The main of them is that \ds \lim_{n\to \infty} \frac{\ln ET(n)}{\ln n+1} exists.Comment: 12 pages, 1 figure, Maple cod