### Triangle Problem

My math teacher showed me this interesting problem. It may seem simple at first but see if you can find a generalization to it.

How many triangles with integer side lengths have perimeter 10?

Alright, this problem isn’t too difficult. But what if we wanted to see how many triangles with integer side lengths have perimeter x for any x? Is there a nice way to generalize this such that the answer is of the form f(x) for some function f? Is there a way to prove that this generalization is true for any integer x?