Asymptotically,
results of Klarner et al. [Kla97, Thm. 12.3.1] show
that the number of fixed n-ominoes (factoring out just translations),
denoted t(n), satisfies
[
t(
n)]
^{1/n} =
λ
(roughly, t(n) is around n^{λ})
for ``Klarner's constant" λ,
with
4.0025 < λ < 4.5685,
but the precise value of λ remains open.
The lower bound of > 4 is established in [BRS15],
and the upper bound in [BB15].
The exact counts have been computed for small n.
See [Sloa] for the number of fixed n-ominoes for n≤28
and for related references.
The current record is n = 56 by Jensen [Jen03].
See also [Epp] for related links.