cp's OEIS Frontend

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.

Showing 1-7 of 7 results.

A360256 Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.

Original entry on oeis.org

1, 1, 33, 513, 14409, 693025, 50447161
Offset: 1

Views

Author

Scott R. Shannon, Feb 17 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct height X width dimensions means that, for example, a 1 X 3 rectangle can be used twice, once in a horizontal (1 X 3) and once in a vertical (3 X 1) direction.

Examples

			a(1) = 1 as the only way to tile a 1 X 1 square is with a square with dimensions 1 X 1.
a(2) = 1 as the only way to tile a 2 X 2 square is with a square with dimensions 2 X 2.
a(3) = 33. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |   |       |   |   |       |   |       |   |   |           |   |   |       |
  +---+---+---+   +---+---+---+   +---+---+   +   +---+---+---+   +---+---+---+
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +   +       +   +           +   +       +   +   +           +   +       +   +
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 4 different ways, the second in 8 different ways, the third in 8 different ways, the fourth in 4 different ways and the fifth in 8 different ways. There is also the single 3 X 3 rectangle. This gives 33 ways in total.

Crossrefs

Cf. A360725 (oblongs), A360499, A360498, A182275, A004003, A099390, A065072.

A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

Original entry on oeis.org

0, 0, 4, 12, 256, 3620, 87216, 2444084, 87181220
Offset: 1

Views

Author

Scott R. Shannon, Feb 09 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 oblong can only be used once, regardless of if it lies horizontally or vertically.

Examples

			a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
.
  +---+---+---+
  |           |
  +---+---+---+
  |           |
  +           +
  |           |
  +---+---+---+
.
This tiling can occur in 4 different ways, giving 4 ways in total.
a(4) = 12. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+   +---+---+---+---+
  |   |           |   |               |
  +   +           +   +---+---+---+---+
  |   |           |   |               |
  +---+---+---+---+   +               +
  |               |   |               |
  +               +   +               +
  |               |   |               |
  +---+---+---+---+   +---+---+---+---+
.
The first tiling can occur in 8 different way and the second in 4 different ways, giving 12 ways in total.

Crossrefs

Cf. A360499 (rectangles), A004003, A099390, A065072, A233320, A230031.

A360725 Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.

Original entry on oeis.org

0, 0, 4, 36, 1056, 31052, 1473944, 87469884
Offset: 1

Views

Author

Scott R. Shannon, Feb 18 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct height x width dimensions means that, for example, a 1 x 3 oblong can be used twice, once in a horizonal (1 x 3) and once in a vertical (3 x 1) direction.

Examples

			a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
.
  +---+---+---+
  |           |
  +---+---+---+
  |           |
  +           +
  |           |
  +---+---+---+
.
This tiling can occur in 4 different ways, giving 4 ways in total.
a(4) = 36. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |           |   |               |   |   |           |   |   |           |
  +   +           +   +---+---+---+---+   +   +---+---+---+   +   +---+---+---+
  |   |           |   |               |   |   |           |   |   |   |       |
  +---+---+---+---+   +               +   +   +           +   +   +   +       +
  |               |   |               |   |   |           |   |   |   |       |
  +               +   +               +   +---+---+---+---+   +---+---+       +
  |               |   |               |   |               |   |       |       |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
The first tiling can occur in 8 different ways, the second in 4 different ways, the third in 16 different ways and the fourth in 8 different ways. This gives 36 ways in total.

Crossrefs

Cf. A360256 (rectangles), A360499, A360498, A182275, A004003, A099390, A065072.

A360773 Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.

Original entry on oeis.org

0, 1, 8, 1024, 620448
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Feb 20 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.
Only squares with even edges lengths are possible since the area of a square with odd edge lengths is odd, while the perimeter of any rectangle is even.

Examples

			a(1) = 0 as a 2 x 2 square, with area 4, cannot be tiled with distinct rectangles with perimeters that sum to 4.
a(2) = 1 as a 4 x 4 rectangle, with area 16, can be tiled with a 4 x 4 square with perimeter 4 + 4 + 4 + 4 = 16.
a(3) = 8. The possible tilings for the 6 x 6 square, with area 36, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+---+---+   +---+---+---+---+---+---+
  |                       |   |                       |
  +---+---+---+---+---+---+   +                       +
  |                       |   |                       |
  +                       +   +---+---+---+---+---+---+
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +---+---+---+---+---+---+   +---+---+---+---+---+---+
.
where for the first tiling (2*6 + 2*1) + (2*6 + 2*5) = 36 while for the second tiling (2*6 + 2*2) + (2*6 + 2*4) = 36. Both of these tilings can occur in 4 ways, giving 8 ways in total.
a(4) = 1024. And example tiling of the 8 x 8 square, with area 64, is:
.
  +---+---+---+---+---+---+---+---+
  |   |                   |       |
  +   +                   +---+---+
  |   |                   |       |
  +   +                   +       +
  |   |                   |       |
  +---+---+---+---+---+---+---+---+
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +---+---+---+---+---+---+---+---+
.
where (2*1 + 2*3) + (2*5 + 2*3) + (2*2 + 2*1) + (2*2 + 2*2) + (2*8 + 2*5) = 64.

Crossrefs

Cf. A360499, A360498, A360725, A360256, A182275, A004003, A099390, A065072.

A360804 Number of ways to tile an n X n square using rectangles with distinct areas.

Original entry on oeis.org

1, 1, 21, 253, 2401, 36237, 815929, 18713197
Offset: 1

Views

Author

Scott R. Shannon, Feb 21 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct areas means that, for example, only one of the two rectangles with area 4, a 2 X 2 or 1 X 4 rectangle, can be used in any tiling.

Examples

			a(1) = 1 as the only way to tile a 1 X 1 square is with a square with dimensions 1 X 1.
a(2) = 1 as the only way to tile a 2 X 2 square is with a square with dimensions 2 X 2.
a(3) = 21. The possible tilings are the same as those given in the examples of A360499(3).
a(4) = 253. And example tiling of the 4 X 4 square is:
.
  +---+---+---+---+
  |   |       |   |
  +---+---+---+   +
  |           |   |
  +           +   +
  |           |   |
  +---+---+---+---+
  |               |
  +---+---+---+---+
.
which contains rectangles with areas 1, 2, 3, 4, 6. The one tiling, excluding symmetrically equivalent arrangements, that is excluded here but allowed in A360499 is:
.
  +---+---+---+---+
  |       |       |
  +       +       +
  |       |       |
  +---+---+       +
  |       |       |
  +---+---+---+---+
  |               |
  +---+---+---+---+
.
as this contains two rectangles with area 4. This can occur in 16 different ways so a(4) = A360499(4) - 16 = 269 - 16 = 253.

Crossrefs

Cf. A360499, A360498, A360725, A360256, A360773, A182275, A004003.

A360943 Number of ways to tile an n X n square using rectangles with distinct dimensions where no rectangle has an edge length that divides n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 360, 0, 360, 360, 8547192, 0
Offset: 1

Views

Author

Scott R. Shannon, Mar 01 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 2 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.
Other known values are a(14) = 517344, a(15) = 6068760, a(16) = 339312. a(13) is greater than 800 million.

Examples

			a(1)..a(6),a(8),a(12) = 0 as these squares cannot be tiled with distinct rectangles with edge lengths that do not divide n. For example for the 8 x 8 square only three rectangles are available with dimensions 3 x 3, 3 x 5, and 5 x 5. All other rectangles have an edge length that divides 8 else leave a space of size 1 or 2 units between its edge and the edge of the square. These gaps cannot be filled as no rectangle can have an edge length of 1 or 2.
a(7) = 360. And example tiling is:
.
  +---+---+---+---+---+---+---+
  |       |           |       |
  +       +           +       +
  |       |           |       |
  +---+---+---+---+---+       +
  |                   |       |
  +                   +       +
  |                   |       |
  +---+---+---+---+---+---+---+
  |           |               |
  +           +               +
  |           |               |
  +           +               +
  |           |               |
  +---+---+---+---+---+---+---+
.

Crossrefs

Cf. A360499, A360804, A360256, A360773, A182275, A004003.

A361413 Number of ways to tile an n X n square using rectangles with distinct dimensions where all the rectangle edge lengths are prime numbers.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 4128, 1, 10880, 641, 45904, 349496, 892088, 40873, 17695080
Offset: 1

Views

Author

Scott R. Shannon, Mar 10 2023

Keywords

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 2 X 3 rectangle can only be used once, regardless of whether it lies horizontally or vertically.

Examples

			a(2), a(3), a(5), a(7), a(11) = 1 as the only possible tiling is that using an n X n square where n is a prime number. It is likely 11 is the last prime indexed term that equals 1 although this is unknown.
a(10) = 4128. And example tiling is:
.
  +---+---+---+---+---+---+---+---+---+---+
  |       |           |                   |
  +       +           +                   +
  |       |           |                   |
  +---+---+---+---+---+---+---+---+---+---+
  |           |                           |
  +           +                           +
  |           |                           |
  +           +                           +
  |           |                           |
  +---+---+---+                           +
  |           |                           |
  +           +                           +
  |           |                           |
  +           +---+---+---+---+---+---+---+
  |           |                           |
  +           +                           +
  |           |                           |
  +           +                           +
  |           |                           |
  +---+---+---+---+---+---+---+---+---+---+
.

Crossrefs

Cf. A360943, A360499, A360804, A360256, A360773, A182275, A004003.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.