A characterization of all equilateral triangles in \Bbb Z^3

TL;DR

This paper extends previous theorems to characterize all equilateral triangles with integer vertices in three-dimensional space, providing an extrapolation formula and analyzing the sequence's asymptotic behavior.

Contribution

It generalizes the construction of integer-coordinate equilateral triangles and offers an approximate formula for their count in a finite cube.

Findings

Extended the theorem for constructing equilateral triangles with integer vertices in 

Provided an extrapolation formula for the sequence ET(n)

Analyzed the asymptotic behavior of ET(n)

Abstract

This paper is a continuation of previous work of the authors. We extend one of the theorems that gave a way to construct equilateral triangles whose vertices have integer coordinates to the general situation. An approximate extrapolation formula for the sequence ET(n) of all equilateral triangles with vertices in $\{0,1,2,...,n\}^3$ (A 102698) is given and the asymptotic behavior of this sequence is analyzed.