I think I found an easier solution for this specific problem without complicating it too much.
We have one basic kind of triangle, and that is the one that our big triangle contains 4 of. Let's call that order 1.
If we count the triangles inside order 1, we find out that they are 16. They are not so many so we can actually count them manually.
So long we have 16 triangles times 4, 64 triangles.
Now by combining 4 order-1 triangles we get new triangles that originate from the combination.
Lets count the new triangles for ONE side only. Be careful because some of them can be mirrored and some cannot. Actually there is only one that cannot be mirrored.
That gives us 13 new triangles for ONE side only from the combination of order 1 into order 2.
Now we have 13 more for every time we turn the big triangle around. So now we have:
16*4 + 13*3 = 103
Don't forget the big triangle!
16*4 + 13*3 + 1 = 104
Woho, solved! =)