Background
A maximal domino placement (MDP) on a rectangular grid is a non-overlapping placement of zero or more dominoes, so that no more dominoes can be added without overlapping some existing domino.
Alternatively, it can be thought of as a tiling using dominoes and monominoes (single square pieces) so that no two monominoes are adjacent to each other.
For example, the following are a few MDPs on a 3x3 grid: (-s and |s represent horizontal and vertical dominoes, and os represent holes respectively.)
--| --| --o|o| --| o--|-- --o --oThere are exactly five MDPs on 2x3, and eleven on 2x4. Rotation and/or reflection of a placement is different from original unless they exactly coincide.
||| |-- --| o-- --o||| |-- --| --o o--|||| ||-- |--| |o-- |--o --|||||| ||-- |--| |--o |o-- --||--o| ---- o--| o--o ----o--| o--o --o| ---- ----In the graph-theoretical sense, an MDP is equivalent to a maximal matching (maximal independent edge set) in the grid graph of given size.
Challenge
Given the width and height of a grid, count the number of distinct maximal domino placements on it.
Standard code-golf rules apply. The shortest code in bytes wins.
Test cases
A288026 is the table of values read by antidiagonals.
w|h| 1 2 3 4 5 6---+------------------------------1 | 1 1 2 2 3 42 | 1 2 5 11 24 513 | 2 5 22 75 264 9414 | 2 11 75 400 2357 134075 | 3 24 264 2357 22228 2074236 | 4 51 941 13407 207423 3136370