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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A361216 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle.

Original entry on oeis.org

1, 1, 4, 2, 11, 56, 3, 29, 370, 5752, 4, 94, 2666, 82310, 2519124, 6, 263, 19126, 1232770, 88117873, 6126859968, 12, 968, 134902, 19119198, 2835424200
Offset: 1

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

Comments

Tilings that are rotations or reflections of each other are considered distinct.
Pieces can have any combination of integer side lengths, but for the optimal sets computed so far (up to (n,k) = (7,5)), all pieces have one side of length 1.

Examples

			Triangle begins:
  n\k|  1    2       3         4            5          6  7  8
  ---+--------------------------------------------------------
  1  |  1
  2  |  1    4
  3  |  2   11      56
  4  |  3   29     370      5752
  5  |  4   94    2666     82310      2519124
  6  |  6  263   19126   1232770     88117873 6126859968
  7  | 12  968  134902  19119198   2835424200          ?  ?
  8  | 20 3416 1026667 307914196 109979838540          ?  ?  ?
A 3 X 3 square can be tiled by three 1 X 2 pieces and three 1 X 1 pieces in the following ways:
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+   +   +---+---+   +---+   +---+
  |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +   +---+---+   +   +---+---+   +
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |       |   |   |   |   |   |   |   |       |
  +---+---+---+   +---+---+   +   +---+---+---+
  |   |   |   |   |       |   |   |       |   |
  +---+---+   +   +---+---+---+   +---+---+---+
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
  |       |   |   |   |       |
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
The first six of these have no symmetries, so they account for 8 tilings each. The last two has a mirror symmetry, so they account for 4 tilings each. In total there are 6*8+2*4 = 56 tilings. This is the maximum for a 3 X 3 square, so T(3,3) = 56.
The following table shows the sets of pieces that give the maximum number of tilings up to (n,k) = (7,5). The solutions are unique except for (n,k) = (2,1) and (n,k) = (6,1).
      \     Number of pieces of size
  (n,k)\  1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
  ------+-------+-------+-------+------
  (1,1) |   1   |   0   |   0   |   0
  (2,1) |   2   |   0   |   0   |   0
  (2,1) |   0   |   1   |   0   |   0
  (2,2) |   2   |   1   |   0   |   0
  (3,1) |   1   |   1   |   0   |   0
  (3,2) |   2   |   2   |   0   |   0
  (3,3) |   3   |   3   |   0   |   0
  (4,1) |   2   |   1   |   0   |   0
  (4,2) |   4   |   2   |   0   |   0
  (4,3) |   3   |   3   |   1   |   0
  (4,4) |   5   |   4   |   1   |   0
  (5,1) |   3   |   1   |   0   |   0
  (5,2) |   4   |   3   |   0   |   0
  (5,3) |   4   |   4   |   1   |   0
  (5,4) |   7   |   5   |   1   |   0
  (5,5) |   7   |   6   |   2   |   0
  (6,1) |   2   |   2   |   0   |   0
  (6,1) |   1   |   1   |   1   |   0
  (6,2) |   4   |   4   |   0   |   0
  (6,3) |   7   |   4   |   1   |   0
  (6,4) |   8   |   5   |   2   |   0
  (6,5) |  10   |   7   |   2   |   0
  (6,6) |  11   |   8   |   3   |   0
  (7,1) |   2   |   1   |   1   |   0
  (7,2) |   5   |   3   |   1   |   0
  (7,3) |   8   |   5   |   1   |   0
  (7,4) |  10   |   6   |   2   |   0
  (7,5) |  11   |   7   |   2   |   1

Crossrefs

Main diagonal: A361217.
Columns: A102462 (k = 1), A361218 (k = 2), A361219 (k = 3), A361220 (k = 4).
Cf. A360629, A361221.

Formula

T(n,1) = A102462(n).

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.

Original entry on oeis.org

1, 1, 5, 12, 31, 86, 242, 854, 2888, 10478, 34264, 120347
Offset: 1

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

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

A361426 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.

Original entry on oeis.org

2, 2, 6, 12, 16, 48, 53, 120, 320, 280, 1120, 2240, 2986, 8960, 17920, 26880, 53760, 107520, 134400, 268800, 537600, 591360, 1182720, 2365440, 2956800, 5677056, 11354112
Offset: 1

Views

Author

Pontus von Brömssen, Mar 11 2023

Keywords

Comments

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).

Examples

			The following table shows all sets of pieces that give the maximum (n,2)-tiling difficulty level up to n = 27.
    \           Number of pieces of size
   n \  1X1 | 1X2 | 1X3 | 1X4 | 1X5 | 1X7 | 2X2 | 2X3
  ----+-----+-----+-----+-----+-----+-----+-----+----
   1  |  0  |  1  |  0  |  0  |  0  |  0  |  0  |  0
   2  |  0  |  2  |  0  |  0  |  0  |  0  |  0  |  0
   3  |  1  |  1  |  1  |  0  |  0  |  0  |  0  |  0
   4  |  0  |  2  |  0  |  1  |  0  |  0  |  0  |  0
   4  |  0  |  1  |  2  |  0  |  0  |  0  |  0  |  0
   5  |  1  |  2  |  0  |  0  |  1  |  0  |  0  |  0
   5  |  0  |  3  |  0  |  1  |  0  |  0  |  0  |  0
   6  |  0  |  1  |  2  |  1  |  0  |  0  |  0  |  0
   7  |  0  |  1  |  4  |  0  |  0  |  0  |  0  |  0
   8  |  2  |  0  |  2  |  1  |  0  |  0  |  1  |  0
   8  |  0  |  1  |  2  |  1  |  0  |  0  |  1  |  0
   9  |  1  |  0  |  3  |  2  |  0  |  0  |  0  |  0
  10  |  2  |  0  |  2  |  1  |  0  |  0  |  2  |  0
  11  |  1  |  0  |  3  |  2  |  0  |  0  |  1  |  0
  12  |  1  |  0  |  3  |  2  |  0  |  0  |  0  |  1
  12  |  0  |  0  |  4  |  3  |  0  |  0  |  0  |  0
  13  |  1  |  0  |  3  |  2  |  0  |  0  |  2  |  0
  14  |  0  |  0  |  4  |  3  |  0  |  0  |  1  |  0
  15  |  0  |  0  |  4  |  3  |  0  |  0  |  0  |  1
  16  |  0  |  0  |  4  |  3  |  0  |  0  |  2  |  0
  17  |  0  |  0  |  4  |  3  |  0  |  0  |  1  |  1
  18  |  0  |  0  |  4  |  3  |  0  |  0  |  0  |  2
  19  |  0  |  0  |  4  |  3  |  0  |  0  |  2  |  1
  20  |  0  |  0  |  4  |  3  |  0  |  0  |  1  |  2
  21  |  0  |  0  |  4  |  3  |  0  |  0  |  0  |  3
  22  |  0  |  0  |  5  |  2  |  0  |  1  |  2  |  1
  22  |  0  |  0  |  5  |  0  |  3  |  0  |  2  |  1
  22  |  0  |  0  |  4  |  3  |  0  |  0  |  2  |  2
  23  |  0  |  0  |  5  |  2  |  0  |  1  |  1  |  2
  23  |  0  |  0  |  5  |  0  |  3  |  0  |  1  |  2
  23  |  0  |  0  |  4  |  3  |  0  |  0  |  1  |  3
  24  |  0  |  0  |  5  |  2  |  0  |  1  |  0  |  3
  24  |  0  |  0  |  5  |  0  |  3  |  0  |  0  |  3
  24  |  0  |  0  |  4  |  3  |  0  |  0  |  0  |  4
  25  |  0  |  0  |  3  |  4  |  0  |  1  |  0  |  3
  26  |  0  |  0  |  5  |  2  |  0  |  1  |  1  |  3
  26  |  0  |  0  |  5  |  0  |  3  |  0  |  1  |  3
  27  |  0  |  0  |  5  |  2  |  0  |  1  |  0  |  4
  27  |  0  |  0  |  5  |  0  |  3  |  0  |  0  |  4

Crossrefs

Second column of A361424.
Cf. A360631, A361218, A361224.
Showing 1-3 of 3 results.

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