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Numerically, it seems like the sum tends towards $\sqrt{N/2}+2^{-3/2}(-1)^{N}$ to pretty good accuracy. I suspect it should be possible to prove something like this using some clever number theoretic argument, maybe. At least a lower bound of the form $a\sqrt{N}$ with $0<a<2^{-1/2}$ should be quite doable, though I'm not seeing how at the moment.
