This is essentially ordering the lattice points intersected by a plane in three space in the all-positive-or-zero octant where the sum of the $x,y,z$-coordinates is $n$.

(I may be mistaken but your example for n=1,2,3 show the values for n=0,1,2, and your example for n=4 does show the answer for 4, so I think you're off by one for part of your example.  And you use the word *permutation*, but your description of the problem is more aptly states as *combination* of three integers.)

This is essentially a simple geometric problem in 3-space over the integers, unless I am misunderstanding your question.

If you look at it geometrically, you are looking at the points on the lattice $\mathbb {Z}^3$
 and finding the points on the plane $x+y+z=n$ in the all positive-or-zero-octant $x\ge0, y\ge0, z\ge0$.  You can then order the lattice points which satisfy these constraints ordinally in whatever order you prefer, say numerically with $x$-coordinate taking precedence over $y$ taking precedence over $z$.

In this case, it is easy to read off the coordinates of the triangle in the all-positive octant of 3-space.

    for x = 0 to n
    for y = 0 to n-x
    z = n-x-y
    print x,y,z
    next y
    next x