The important fact here is that in a triangle with sides , it must be true that and

The last two inequalities are obvious given that , so we just need to see how many ways we can choose and so that

Suppose is even.

Then if , then must equal .

if , then can equal or .

..

if , then b can equal .

(So far we have ways to choose and .

Doing the same thing when , we find that there are more ways to choose, giving a total of triangles.

The other cases of this problem are not much different.