math - Generating triangular/hexagonal coordinates (xyz)

Question

I'm trying to come up with an iterative function that generates xyz coordinates for a hexagonal grid. With a starting hex position (say 0,0,0 for simplicity), I want to calculate the coordinates for each successive "ring" of hexagons, as illustrated here:

So far, all I've managed to come up with is this (example in javascript):

var radius = 3
var xyz = [0,0,0];

// for each ring
for (var i = 0; i < radius; i++) {
    var tpRing = i*6;
    var tpVect = tpRing/3;
    // for each vector of ring
    for (var j = 0; j < 3; j++) {
        // for each tile in vector
        for(var k = 0; k < tpVect; k++) {
            xyz[0] = ???;
            xyz[1] = ???;
            xyz[2] = ???;
            console.log(xyz);
        }
    }
}

I know each ring contains six more points than the previous and each 120° vector contains one additional point for each step from the center. I also know that x + y + z = 0 for all tiles. But how can I generate a list of coordinates that follow the sequence below?

    0, 0, 0

    0,-1, 1
    1,-1, 0
    1, 0,-1
    0, 1,-1
   -1, 1, 0
   -1, 0, 1

    0,-2, 2
    1,-2, 1
    2,-2, 0
    2,-1,-1
    2, 0,-2
    1, 1,-2
    0, 2,-2
   -1, 2,-1
   -2, 2, 0
   -2, 1, 1
   -2, 0, 2
   -1,-1, 2

Answer 1

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("");
}