# A forest of Eisensteinian triplets

## TL;DR

This paper extends Berggren's matrix product characterization from Pythagorean triplets to Eisensteinian triplets, showing that all such triangles can be generated by multiplying specific base triplets with a sequence of five matrices.

## Contribution

It introduces a matrix product framework for generating Eisensteinian triplets, analogous to Berggren's method for Pythagorean triplets, providing a new tool for their enumeration.

## Key findings

- All Eisensteinian triplets can be generated from base triplets using five matrices.
- The method may help enumerate points with rational distances on a hexagonal lattice.
- The approach parallels Berggren's matrix product for Pythagorean triplets.

## Abstract

In 1934 B. Berggren first discovered the surprising result that every Pythagorean triplet is the pre product of the triplet (3, 4, 5) presented as a column by a product of three matrices, that every triplet is obtained in this manner exactly once and in primitive form. In this paper we show a similar result for integer triangles with an angle of 60 degrees (also known as Eisensteinian triplets). We show that any such triangle is obtained by pre-multiplication (7,8,5) or (13,15,7) by a product of five matrices. The result might have applications in enumerating points with rational distance from the origin on the hexagonal lattice.