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

1,3

Links

Table of n, a(n) for n=1..12.

Example

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:
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +---+---+   +---+---+   +---+---+   +   +---+   +---+---+   +---+---+
  |   |   |   |   |   |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +   +---+   +---+---+   +---+---+   +---+---+   +---+---+
  |       |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +   +   +
  |       |   |       |   |       |   |       |   |       |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
.
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +   +---+   +---+   +   +---+---+   +---+---+
  |   |   |   |       |   |   |   |   |   |   |   |       |   |   |   |
  +---+   +   +---+---+   +---+---+   +---+---+   +---+---+   +   +   +
  |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +   +---+   +   +---+   +   +---+   +   +---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
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.

Crossrefs

Second column of A361221.

Cf. A361218, A360631.

Sequence in context: A038357 A090974 A038606 * A327323 A192303 A301785

Adjacent sequences: A361221 A361222 A361223 * A361225 A361226 A361227

Keyword

nonn,more

Author

Pontus von Brömssen, Mar 05 2023

Status

approved