92
xkcd #3125: Snake-in-the-Box Problem
(imgs.xkcd.com)
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I'd expect something around ~200 for n=9 and ~400 for n=10, but I imagine this is too big to be brute forced by raw computing
Some lower bounds have been established: https://oeis.org/A099155
Some trivial bounds: F(n-1) + 1 <= F(n) <= F(n-1) * 2 + 1.
Also F(n) <= 2^(n-1)