Counting all regular tetrahedra in {0,1,...,n}^3

TL;DR

This paper presents a method to count all regular tetrahedra with vertices in the integer grid {0,1,...,n}^3, contributing to understanding their enumeration and providing tools for estimating their total count.

Contribution

It introduces a procedure for counting all such tetrahedra and develops results to aid in estimating their total number, advancing combinatorial geometry research.

Findings

Developed a counting procedure for regular tetrahedra in integer grids

Derived results to estimate the sequence of tetrahedra counts

Connected the sequence to OEIS entry A103158

Abstract

In this note we describe a procedure of calculating the number all regular tetrahedra that have coordinates in the set {0,1,...,n}. We develop a few results that may help in finding good estimates for this sequence which is twice A103158 in the Online Encyclopedia of Integer Sequences.