A360631 -id:A360631 - cp's OEIS frontend

A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n.

1, 2, 4, 3, 10, 21, 5, 22, 73, 192, 7, 44, 190, 703, 2035, 11, 91, 510, 2287, 8581, 27407, 15, 172, 1196, 6738, 30209, 118461, 399618, 22, 326, 2895, 19160, 102092, 462114

Offset: 1

Pontus von Brömssen, Feb 14 2023

Pieces are free to rotate by 90 degrees, i.e., an r X s piece and an s X r piece are equivalent. See A360451 for the case when the pieces are fixed.

			Triangle begins:
   n\k|  1   2    3    4     5      6      7
   ---+--------------------------------------
   1  |  1
   2  |  2   4
   3  |  3  10   21
   4  |  5  22   73  192
   5  |  7  44  190  703  2035
   6  | 11  91  510 2287  8581  27407
   7  | 15 172 1196 6738 30209 118461 399618
   ...
T(2,2) = 4, because a 2 X 2 rectangle can be tiled by: one 2 X 2 piece; two 1 X 2 pieces; one 1 X 2 piece and two 1 X 1 pieces; four 1 X 1 pieces.
The T(3,2) = 10 sets of pieces that can tile a 3 X 2 rectangle are shown in the table below. (Each column on the right gives a set of pieces.)
   length X width |  number of pieces
   ---------------+--------------------
        2 X 3     | 1 0 0 0 0 0 0 0 0 0
        2 X 2     | 0 1 1 0 0 0 0 0 0 0
        1 X 3     | 0 0 0 2 1 1 0 0 0 0
        1 X 2     | 0 1 0 0 1 0 3 2 1 0
        1 X 1     | 0 0 2 0 1 3 0 2 4 6

Cf. A000041 (column k=1), A116694, A224697 (square pieces), A360451 (fixed pieces), A360630 (main diagonal), A360631 (column k=2), A360632 (column k=3).

T(7,7) and T(8,k) for k = 1..6 added by Robin Visser, May 09 2025

A360632 Number of sets of integer-sided rectangular pieces that can tile a 3 X n rectangle.

Original entry on oeis.org

1, 3, 10, 21, 73, 190, 510, 1196, 2895, 6437, 14281, 29840, 62405, 124506, 246383, 473094, 899000, 1665763, 3057894, 5500889, 9808150

Offset: 0

Pontus von Brömssen, Feb 14 2023

			From _Robin Visser_, May 01 2025: (Start)
For n = 1, there are a(1) = 3 possible sets of rectangular pieces that can tile a 3 x 1 rectangle: one 1 x 3 piece; one 1 x 2 piece and one 1 x 1 piece; or three 1 x 1 pieces.
For n = 2, there are a(2) = 10 possible sets of rectangular pieces that can tile a 3 x 2 rectangle: one 2 x 3 piece; one 2 x 2 piece and one 1 x 2 piece; one 2 x 2 piece and two 1 x 1 pieces; two 1 x 3 pieces; one 1 x 3 piece, one 1 x 2 piece, and one 1 x 1 piece; one 1 x 3 piece and three 1 x 1 pieces; three 1 x 2 pieces; two 1 x 2 pieces and two 1 x 1 pieces; one 1 x 2 piece and four 1 x 1 pieces; or six 1 x 1 pieces. (End)

Third column of A360629.

Cf. A000041 (1 x n rectangle), A360631 (2 x n rectangle).

def a(n):
    A = [[[set() for i in range(n+1)] for j in range(n+1)] for k in range(n+1)]
    A[0][0][0].add(()); m = n+1;
    for (i,j,k) in [(x,y,z) for x in range(m) for y in range(m) for z in range(m)]:
        for (l,p) in [(x,y) for x in range(1,i+1) for y in A[i-x][j][k]]:
            A[i][j][k].add(tuple(sorted(list(p)+[(1,l)])))
        for (l,p) in [(x,y) for x in range(1,j+1) for y in A[i][j-x][k]]:
            A[i][j][k].add(tuple(sorted(list(p)+[(1,l)])))
        for (l,p) in [(x,y) for x in range(1,k+1) for y in A[i][j][k-x]]:
            A[i][j][k].add(tuple(sorted(list(p)+[(1,l)])))
        for (l,p) in [(x,y) for x in range(1,min(i,j)+1) for y in A[i-x][j-x][k]]:
            if (i==j): A[i][j][k].add(tuple(sorted(list(p)+[tuple(sorted((2,l)))])))
        for (l,p) in [(x,y) for x in range(1,min(j,k)+1) for y in A[i][j-x][k-x]]:
            if (j==k): A[i][j][k].add(tuple(sorted(list(p)+[tuple(sorted((2,l)))])))
        for (l,p) in [(x,y) for x in range(1,min(i,j,k)+1) for y in A[i-x][j-x][k-x]]:
            if (i==j==k): A[i][j][k].add(tuple(sorted(list(p)+[tuple(sorted((3,l)))])))
    return len(A[n][n][n])  # Robin Visser, May 01 2025

a(17)-a(20) from Robin Visser, May 04 2025

a(0) = 1 prepended by Robin Visser, May 05 2025

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

Pontus von Brömssen, Mar 05 2023

			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.

Second column of A361221.

Cf. A361218, A360631.

A361218 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 2 rectangle.

Original entry on oeis.org

1, 4, 11, 29, 94, 263, 968, 3416, 11520, 41912, 136972, 481388, 1743784, 6275886, 23615432, 93819128, 368019576, 1367900808, 5403282616, 19831367476, 76031433360, 300581321056, 1143307393600, 4542840116352, 17001097572544, 65314285778004, 246695766031432

Offset: 1

Pontus von Brömssen, Mar 05 2023

nonn

Tilings that are rotations or reflections of each other are considered distinct.

			The following table shows the sets of pieces that give the maximum number of tilings for n <= 27. The solutions are unique except for n = 1.
    \     Number of pieces of size
   n \  1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
  ----+-------+-------+-------+------
   1  |   2   |   0   |   0   |   0
   1  |   0   |   1   |   0   |   0
   2  |   2   |   1   |   0   |   0
   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
  13  |   8   |   6   |   2   |   0
  14  |  10   |   6   |   2   |   0
  15  |  10   |   7   |   2   |   0
  16  |  10   |   6   |   2   |   1
  17  |  10   |   7   |   2   |   1
  18  |  12   |   7   |   2   |   1
  19  |  12   |   8   |   2   |   1
  20  |  12   |   9   |   2   |   1
  21  |  13   |   8   |   3   |   1
  22  |  13   |   9   |   3   |   1
  23  |  15   |   9   |   3   |   1
  24  |  15   |  10   |   3   |   1
  25  |  15   |  11   |   3   |   1
  26  |  17   |  11   |   3   |   1
  27  |  17   |  12   |   3   |   1

Second column of A361216.

Cf. A360631, A361224.

A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Original entry on oeis.org

2, 2, 3, 4, 3, 6, 3, 6, 5, 6, 3, 10, 3, 6, 7, 8, 3, 10, 3, 10, 7, 6, 3, 14, 5, 6, 7, 10, 3, 14, 3, 10, 7, 6, 7, 16, 3, 6, 7, 14, 3, 14, 3, 10, 11, 6, 3, 18, 5, 10, 7, 10, 3, 14, 7, 14, 7, 6, 3, 22, 3, 6, 11, 12, 7, 14, 3, 10, 7, 14, 3, 22, 3, 6, 11, 10, 7, 14

Offset: 1

Pontus von Brömssen, Feb 28 2023

nonn

Second column of A360998.
Essentially the same as A086369.
Cf. A000005, A000035, A114003, A360631, A361004.

a(n) = 2*A000005(n) - 1 - [n even] = A114003(n) + A000035(n) - 1 for n >= 2.

A361004 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling).

Original entry on oeis.org

2, 4, 7, 11, 14, 23, 22, 32, 37, 42, 39, 69, 50, 64

Offset: 1

Pontus von Brömssen, Feb 28 2023

Second column of A361001.

Cf. A360631, A360999.

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

Pontus von Brömssen, Mar 11 2023

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).

			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

Second column of A361424.

Cf. A360631, A361218, A361224.

cp's OEIS Frontend

A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A360632 Number of sets of integer-sided rectangular pieces that can tile a 3 X n rectangle.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Extensions

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.

Views

Author

Keywords

Examples

Crossrefs

A361218 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 2 rectangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Views

Author

Keywords

Crossrefs

Formula

A361004 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling).

Views

Author

Keywords

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs