In any golygon, all horizontal edges have the same parity as each other, as do all vertical edges. Therefore, the number n of sides must allow the solution of the system of equations It follows from this that n must be a multiple of 8. The number of solutions to this system of equations may be computed efficiently using generating functions (sequence A007219 in OEIS) but finding the number of solutions that correspond to non-crossing golygons seems to be significantly more difficult.
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