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.
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.
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