  We provide an algorithm for computing the number of integral points lying in
certain triangles that do not have integral vertices. We use techniques from
Algebraic Geometry such as the Riemann-Roch formula for weighted projective
planes and resolution of singularities. We analyze the complexity of the method
and show that the worst case is given by the Fibonacci sequence. At the end of
the manuscript a concrete example is developed in detail where the interplay
with other invariants of singularity theory is also treated.
