Number of integer triangles with sides 4 or less 
Consider triangles having integer sides such that no side is greater than 4 units. How many such triangles are possible?

I suspect a relation to the following question:

How many ways can $r$ things be taken from $n$ with repetition and without regard to order?


In general, the number $a(n)$ of triangles with positive integer side-lengths of at most $n$ can be computed recursively by $a(n) = a(n-2) + {n+1 \choose 2}$. See OEIS sequence A2623 for more info.
Clearly $a(0)=0$. Then $a(2) = a(0) + {3 \choose 2} = 3$ and $a(4) = a(2) + {5 \choose 2} = 13$. See астон's answer for a case analysis that gives you $13$.
