Counting triangles with integer perimeter
Jelly, 3 bytes

«Ḥ’


Try it online!

Explanation

«   min(a,b)
 Ḥ          *2
  ’           -1


(Not quite rigorous) proof of correctness:


Let x = min(a,b) and y = max(a,b), and let z be a potential third side.
We can't have z ≤ y-x, because for z &lt; y-x the triangle can't be closed and for z = y-x it would be degenerate. See triangle inequality.
We can't have z ≥ y+x, for the same reasons.
Every integer z in between is valid and gives a different triangle. That means we've got (y+x) - (y-x) - 1 = 2x - 1 different triangles.