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Maximum difficulty level (see A361424 for the definition) for tiling an n X 2 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.
1

%I #5 Mar 13 2023 13:30:57

%S 2,2,6,12,16,48,53,120,320,280,1120,2240,2986,8960,17920,26880,53760,

%T 107520,134400,268800,537600,591360,1182720,2365440,2956800,5677056,

%U 11354112

%N Maximum difficulty level (see A361424 for the definition) for tiling an n X 2 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.

%C The only cases, currently known to the author, for which the maximum difficulty level is not an integer, are n = 7 (difficulty level 160/3) and n = 13 (difficulty level 8960/3).

%e The following table shows all sets of pieces that give the maximum (n,2)-tiling difficulty level up to n = 27.

%e \ Number of pieces of size

%e n \ 1X1 | 1X2 | 1X3 | 1X4 | 1X5 | 1X7 | 2X2 | 2X3

%e ----+-----+-----+-----+-----+-----+-----+-----+----

%e 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0

%e 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0

%e 3 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0

%e 4 | 0 | 2 | 0 | 1 | 0 | 0 | 0 | 0

%e 4 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 0

%e 5 | 1 | 2 | 0 | 0 | 1 | 0 | 0 | 0

%e 5 | 0 | 3 | 0 | 1 | 0 | 0 | 0 | 0

%e 6 | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 0

%e 7 | 0 | 1 | 4 | 0 | 0 | 0 | 0 | 0

%e 8 | 2 | 0 | 2 | 1 | 0 | 0 | 1 | 0

%e 8 | 0 | 1 | 2 | 1 | 0 | 0 | 1 | 0

%e 9 | 1 | 0 | 3 | 2 | 0 | 0 | 0 | 0

%e 10 | 2 | 0 | 2 | 1 | 0 | 0 | 2 | 0

%e 11 | 1 | 0 | 3 | 2 | 0 | 0 | 1 | 0

%e 12 | 1 | 0 | 3 | 2 | 0 | 0 | 0 | 1

%e 12 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 0

%e 13 | 1 | 0 | 3 | 2 | 0 | 0 | 2 | 0

%e 14 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 0

%e 15 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 1

%e 16 | 0 | 0 | 4 | 3 | 0 | 0 | 2 | 0

%e 17 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 1

%e 18 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 2

%e 19 | 0 | 0 | 4 | 3 | 0 | 0 | 2 | 1

%e 20 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 2

%e 21 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 3

%e 22 | 0 | 0 | 5 | 2 | 0 | 1 | 2 | 1

%e 22 | 0 | 0 | 5 | 0 | 3 | 0 | 2 | 1

%e 22 | 0 | 0 | 4 | 3 | 0 | 0 | 2 | 2

%e 23 | 0 | 0 | 5 | 2 | 0 | 1 | 1 | 2

%e 23 | 0 | 0 | 5 | 0 | 3 | 0 | 1 | 2

%e 23 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 3

%e 24 | 0 | 0 | 5 | 2 | 0 | 1 | 0 | 3

%e 24 | 0 | 0 | 5 | 0 | 3 | 0 | 0 | 3

%e 24 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 4

%e 25 | 0 | 0 | 3 | 4 | 0 | 1 | 0 | 3

%e 26 | 0 | 0 | 5 | 2 | 0 | 1 | 1 | 3

%e 26 | 0 | 0 | 5 | 0 | 3 | 0 | 1 | 3

%e 27 | 0 | 0 | 5 | 2 | 0 | 1 | 0 | 4

%e 27 | 0 | 0 | 5 | 0 | 3 | 0 | 0 | 4

%Y Second column of A361424.

%Y Cf. A360631, A361218, A361224.

%K nonn,more

%O 1,1

%A _Pontus von Brömssen_, Mar 11 2023