Markov integers

A Markov number or Markoff number is a positive integer x, y or zthat is part of a solution to the Markov Diophantine equation

${\displaystyle x^{2}+y^{2}+z^{2}=3xyz,\,}$

studied by Andrey Markoff (1879, 1880).

This is  combinatorial problem.  How much sequence space do I need?  Or, how many indices do I need to interleave three sequences with x,y,z symbols each.  The problem defines a boundary condition, each of the x,y,x must be organized in triples, at least three times and exactly match index count for each of the independent set combined twice with itself.  Why?  It makes an algebra, I can go through a reflex point and generate the nth solution from the nth-1 and nth-2.  It has recursive integer solution, so we have separation between solutions and we can consider approximations that hold the condition within some round off.