A polyomino of size \$n\$ is a contiguous shape made from joining \$n\$ unit squares side by side. A domino is a size-2 polyomino.
A polydomino of size \$2n\$ is defined as a polyomino of size \$2n\$ which can be tiled with \$n\$ dominoes.
The following are some examples of polydominoes for \$n=3\$ (hexominoes).
. O O | . O . . | O O O . | O O O O O O O O . | O O O O | O . O O | O O . | O . . . | |
The following are some hexominoes that are not polydominoes.
. O . | O . . . | O . O . O O O | O O O O | O O O O O O . | O . . . |
Given a positive integer \$n\$, count the number of distinct polydominoes of size \$2n\$. Rotations and reflections are considered as the same shape. Different tilings of the same shape does not count either. You may take the value of \$2n\$ as input instead (you may assume the input is always even in this case).
The following shape has two ways to tile with dominoes, but it is counted only once just like other hexominoes.
. O O O O . O O .
The sequence is OEIS 056786.
The following are the expected results for \$n=1 \cdots 9\$.
1, 4, 23, 211, 2227, 25824, 310242, 3818983, 47752136