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