A361224 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 2 rectangle, up to rotations and reflections.
1, 1, 5, 12, 31, 86, 242, 854, 2888, 10478, 34264, 120347
Offset: 1
Examples
A 4 X 2 rectangle can be tiled by two 1 X 2 pieces and four 1 X 1 pieces in the following 12 ways:
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
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+---+---+ + +---+ +---+---+ +---+---+ +---+---+ +---+---+
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+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ + + +
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.
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+---+---+ +---+---+ + +---+ +---+ + +---+---+ +---+---+
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+---+ + +---+---+ +---+---+ +---+---+ +---+---+ + + +
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+ +---+ + +---+ + +---+ + +---+ +---+---+ +---+---+
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This is the maximum for a 4 X 2 rectangle, so a(4) = 12.
The following table shows the sets of pieces that give the maximum number of tilings for n <= 12. The solutions are unique except for n <= 2.
\ Number of pieces of size
n \ 1 X 1 | 1 X 2 | 1 X 3 | 2 X 2
----+-------+-------+-------+------
1 | 2 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0
2 | 4 | 0 | 0 | 0
2 | 2 | 1 | 0 | 0
2 | 0 | 2 | 0 | 0
2 | 0 | 0 | 0 | 1
3 | 2 | 2 | 0 | 0
4 | 4 | 2 | 0 | 0
5 | 4 | 3 | 0 | 0
6 | 4 | 4 | 0 | 0
7 | 5 | 3 | 1 | 0
8 | 5 | 4 | 1 | 0
9 | 7 | 4 | 1 | 0
10 | 7 | 5 | 1 | 0
11 | 7 | 6 | 1 | 0
12 | 9 | 6 | 1 | 0
It seems that all optimal solutions for A361218 are also optimal here, but for n = 2 there are other optimal solutions.