How many distinct

`n`-

square polyominoes are there? This is actually an

open problem of

statistical mechanics!

For concreteness, we'll consider polyominoes distinct if they're (nonsymmetrical) reflections; this can only throw our count off by a factor of 2, of course. These are also called "animals".

There are exponentially many `n`-square polyominoes. Indeed, if we pick a square and proceed (for `n`-1 steps) by picking either the next square *up* or the next square *right*, we'll get 2^{n-1} polyominoes that are distinct, except maybe they can be rotated onto each other. So there are at least 2^{n-3} polyominoes -- exponentially many.

On the other hand, note that every self-avoiding walk on the lattice has a polyomino as its image, so there are no more than 4×3^{n-1} polyominoes.

We'd like to say there are Θ(`b`^{n}) polyominoes; what is `b`?