This is a nice one: http://oeis.org/A227133 (see also http://oeisf.org/Poster15a.pdf)
I built a straightforward SAT encoding and found a(11) >= 72 because of
. . X X X X X X X . X
X X X . X . X . . X X
X . X X . . . X X X .
X X . . X . . . X . X
. X X X . X . . . X X
. X . . X X X X X X .
X X X . . X . X . X X
. X . X X X . . . X .
X X . X . . X X X . X
X . . X . . X . X X X
X . X X X X X X . . X
and a(12) >= 84 by
. X X X X . X . X . X X
X X . X . . X X X X . .
X . . X . X X . . . X .
X X . . . . X . . X X X
. X X X X . . X . X . X
X . X . X X X X X X X .
X . X X . X . X . . X X
. X X . . X X . . X . X
X X . . X X . . . X X X
. X . X . X X X X . . X
X X X X X . X . X . X .
X . . X . . . X X X X X
but I have no idea whether these configurations are maximal.
What is the asymptotic growth?