Inspired by this OEIS entry.
Background
A saturated domino covering is a placement of dominoes over an area such that
- the dominoes are completely inside the area,
- the dominoes entirely cover the given area,
- the dominoes may overlap, and
- removal of any domino reveals an uncovered cell (thus failing to satisfy condition 2).
The following is an example of a maximal such covering of 3 × 3 rectangle (since dominoes may overlap, each domino is drawn separately):
AA. B.. ..C ... ... ...... B.. ..C .D. ... ...... ... ... .D. EE. .FFChallenge
Given the dimensions (width and height) of a rectangle, compute the maximum number of dominoes in its saturated domino covering.
You can assume the input is valid: the width and height are positive integers, and 1 × 1 will not be given as input.
Standard code-golf rules apply. The shortest code in bytes wins.
Test cases
A193764 gives the answers for square boards. The following test cases were verified with this Python + Z3 code (not supported on TIO).
Only the test cases for n <= m are shown for brevity, but your code should not assume so; it should give the same answer for n and m swapped.
n m => answer1 2 => 11 3 => 21 9 => 61 10 => 62 2 => 22 3 => 42 5 => 73 3 => 63 4 => 83 7 => 154 4 => 124 7 => 21