Not only is x + y + z = 0, but the absolute values of x, y and z are equal to twice the radius of the ring. This should be sufficient to identify every hexagon on each successive ring:
var radius = 4;
for(var i = 0; i < radius; i++)
{
for(var j = -i; j <= i; j++)
for(var k = -i; k <= i; k++)
for(var l = -i; l <= i; l++)
if(Math.abs(j) + Math.abs(k) + Math.abs(l) == i*2 && j + k + l == 0)
console.log(j + "," + k + "," + l);
console.log("");
}