Is there a nice closed-form for this?
Since a straight line intercepts a parabola in at most two points, I believe your looking for the number of lattice points on both the curve and straight line that surround the area between the two lines. Is that right? Otherwise the answer would be one of either 0,1,2 or infinity. Infinity would be if you include all lattice points on the parabola, not just those bordering the area. Then again you may be looking for the number of coordinite points within the area so I am unsure what you are looking for. Also, should we assume that a and b are integers?
The integral lattice points in the area formed between the two curves
Solve the Quadratic equation x^2 - ax - b = 0 to get x small and x large. Then from the ceiling of x small to the floor of x large, sum ax+b +1 - x^2 for integer values of x. I think that will give you your sum.
Final Edit: the above includes the lattice points of the lines, to exclude them, sum the value "ax + b -1 - x^2" for integer values between x-small and x-large. Both sums assume that "a" and "b" are integers.
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