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What I would do, if you really wanted to do this semi-symbolically, is to use Piecewise to create a piecewise function that does $g(x)$ as far out as you need it to go. I guess to plot it out to $n=5$ and $x=1$ would require you to have $g$ out to $2^{32}$... hmm, that seems pretty far!

So you could do this symbolically in Sage, but I don't know that it's actually practical. Probably asking to speed it up would be better. Unfortunately, as you know, piecewise support isn't so hot. It would be great to have piecewise + periodic... sigh.