A219924 -id:A219924 - cp's OEIS frontend

A224861 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 2 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

0, 0, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 3, 3, 15, 0, 0, 4, 9, 38, 75, 0, 0, 9, 9, 68, 77, 604, 0, 0, 13, 21, 160, 311, 2384, 4556

Offset: 1

Christopher Hunt Gribble, Jul 22 2013

			The triangle is:
n\k  1   2   3   4   5   6   7    8 ...
.
0    0   0   0   0   0   0   0    0 ...
1        0   0   0   0   0   0    0 ...
2            1   1   3   4   9   13 ...
3                4   3   9   9   21 ...
4                   15  38  68  160 ...
5                       75  77  311 ...
6                          604 2384 ...
7                              4556 ...
...
T(3,5) = 3 because there are 3 different sets of 2 tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will transform each tiling in a set into the other in the same set.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
An example of a tiling in each set is:
._________.    ._________.    ._________.
|   |_|   |    |   |_|_|_|    |     |_|_|
|_ _|_|_ _|    |___|_|   |    |     |_|_|
|_|_|_|_|_|    |_|_|_|___|    |_____|_|_|

Christopher Hunt Gribble, C++ program

Cf. A219924, A224697, A227690.

A224850(n,k) + T(n,k) + A224867(n,k) = A227690(n,k).

1*A224850(n,k) + 2*T(n,k) + 4*A224867(n,k) = A219924(n,k).

A224867 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 4 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 21, 0, 0, 0, 10, 65, 440, 0, 0, 0, 27, 222, 1901, 14508, 0, 0, 0, 58, 676, 7716, 81119, 856559

Offset: 1

Christopher Hunt Gribble, Jul 22 2013

			The triangle is:
n\k    1      2      3      4      5      6      7      8 ...
.
0      0      0      0      0      0      0      0      0 ...
1             0      0      0      0      0      0      0 ...
2                    0      0      0      0      0      0 ...
3                           1      5     10     27     58 ...
4                                 21     65    222    676 ...
5                                       440   1901   7716 ...
6                                            14508  81119 ...
7                                                  856559 ...
...
T(3,5) = 5 because there are 5 different sets of 4 tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will transform each tiling in a set into another in the same set.  Group  D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
An example of a tiling in each set is:
._________.  ._________.  ._________.  ._________.  ._________.
|   |_|_|_|  |_|   |_|_|  |   |   |_|  |   |_|_|_|  |   |     |
|_ _|_|_|_|  |_|_ _|_|_|  |_ _|_ _|_|  |___|   |_|  |___|     |
|_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|___|_|  |_|_|_____|

Christopher Hunt Gribble, C++ program

Cf. A219924, A224697, A227690.

A224850(n,k) + A224861(n,k) + T(n,k) = A227690(n,k).

1*A224850(n,k) + 2*A224861(n,k) + 4*T(n,k) = A219924(n,k).

A160911 a(n) is the number of arrangements of n square tiles with coprime sides in a rectangular frame, counting reflected, rotated or rearranged tilings only once.

Original entry on oeis.org

1, 1, 2, 5, 11, 29, 84, 267, 921, 3481, 14322, 62306, 285845, 1362662, 6681508, 33483830

Offset: 1

Kevin Johnston, Feb 11 2016

There is only one arrangement of 1 square tile: a 1 X 1 rectangle. There is also only 1 arrangement of 2 square tiles: a 2 X 1 rectangle. There are 2 arrangements of 3 square tiles: a 3 X 1 rectangle (three 1 X 1 tiles) and a 3 X 2 rectangle (a 2 X 2 tile and two 1 X 1 tiles).
Short notation for the 2 possible 3-tile solutions:
  3 X 1: 1,1,1
  3 X 2: 2,1,1
More examples see below.
The smallest tile is not always a unit tile, e.g., one of the solutions for 5 tiles is: 6 X 5: 3,3,2,2,2.
My definition of a unique solution is the "signature" string in this notation: the rectangle size for nonsquares and the list of coprime tile sizes sorted largest to smallest. Rotations and reflections of a known solution are not new solutions; rearrangements of the same size tiles within the same overall boundary are not new solutions. But reorganizations of the same size tiles in different boundaries are unique solutions, such as 4 X 1: 1,1,1,1 and 2 X 2: 1,1,1,1.
From Rainer Rosenthal, Dec 23 2022: (Start)
The above description can be abbreviated as follows:
a(n) is the number of (2+n)-tuples (p X q: t_1,...,t_n) of positive integers, such that:
0. p >= q.
1. gcd(t_1,...,t_n) = 1 and t_i >= t_j for i < j and Sum_{i=1..n} t_i^2 = p * q.
2. Any p X q matrix is the disjoint union of contiguous t_i X t_i minors, i = 1..n. (For contiguous minors resp. submatrices see comments in A350237.)
.
The rectangle size p X q may have gcd(p,q) > 1, as seen in the examples for 3 X 2 and 6 X 4. Therefore a(n) >= A210517(n) for all n, and a(6) > A210517(6).
(End)

			From _Rainer Rosenthal_, Dec 24 2022, updated May 09 2024: (Start)
.
                                 |A|
     |A B|                       |B|
     |C D|  (2 X 2: 1,1,1,1)     |C|    (4 X 1: 1,1,1,1)
                                 |D|
.
                                 |A A|
    |A A A|                      |A A|
    |A A A|                      |B B|
    |A A A| (4 X 3: 3,1,1,1)     |B B|  (5 X 2: 2,2,1,1)
    |B C D|                      |C D|
.
    |A A A|
    |A A A|  <=================   3 X 3 minor A
    |A A A|                       2 X 2 minor B
    |B B C|  (5 X 3: 3,2,1,1)     1 X 1 minor C
    |B B D|                       1 X 1 minor D
  ________________________________________________________
       a(4) = 5 illustrated as (p X q: t_1,t_2,t_3,t_4)
         and as p X q matrices with t_i X t_i minors
.
Example configurations for a(6) = 29:
.
                                    |A A A A|
                                    |A A A A|
                                    |A A A A|
      |A A B|         |A B|         |A A A A|
      |A A C|         |C D|         |B B C D|
      |D E F|         |E F|         |B B E F|
   ______________________________________________
      (3 X 3:        (3 X 2:         (6 X 4:
    2,1,1,1,1,1)   1,1,1,1,1,1)    4,2,1,1,1,1)
.                                       _________________________
      |A A A A A A B B B B B B B|      |           |             |
      |A A A A A A B B B B B B B|      |           |             |
      |A A A A A A B B B B B B B|      |    6      |             |
      |A A A A A A B B B B B B B|      |           |      7      |
      |A A A A A A B B B B B B B|      |           |             |
      |A A A A A A B B B B B B B|      |___________|             |
      |C C C C C D B B B B B B B|      |         |1|_____________|
      |C C C C C E E E E F F F F|      |         |       |       |
      |C C C C C E E E E F F F F|      |    5    |  4    |  4    |
      |C C C C C E E E E F F F F|      |         |       |       |
      |C C C C C E E E E F F F F|      |_________|_______|_______|
     _____________________________    _____________________________
         (13 X 11: 7,6,5,4,4,1)           (13 X 11: 7,6,5,4,4,1)
         [rotated by 90 degrees]         [alternate visualization]
.(End)

See A002839 and A217156 for further references and links.

Stuart E. Anderson, Perfect Squared Rectangles and Squared Squares

Cf. A002839, A005670, A113881, A210517, A217156, A219924, A221843, A221844, A221845, A340726, A342558, A350237.

a(15)-a(16) from Kevin Johnston, Feb 11 2016

Title changed from Rainer Rosenthal, Dec 28 2022

A228267 Number T(n,k,r) of dissections of an n X k X r rectangular cuboid into integer-sided cubes including rotations and reflections; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 5, 10, 1, 1, 5, 1, 11, 31, 1, 35, 167, 2098, 1, 1, 8, 1, 21, 76, 1, 93, 635, 15511, 1, 314, 3354, 185473, 4006722, 1, 1, 13, 1, 43, 210, 1, 269, 2887, 151378, 1, 1213, 22478, 3243515, 143662050, 1, 6427, 235150, 112411358

Offset: 1

Christopher Hunt Gribble, Aug 19 2013

The main diagonal T(n,n,n) is 1, 2, 10, 2098, 4006722, .... - R. J. Mathar and Rob Pratt, Nov 27 2017

			The irregular triangle begins:
.   r 1      2      3      4 ...
n,k
1,1   1
2,1   1
2,2   1      2
3,1   1
3,2   1      3
3,3   1      5     10
4,1   1
4,2   1      5
4,3   1     11     31
4,4   1     35    167   2098
5,1   1
5,2   1      8
5,3   1     21     76
5,4   1     93    635  15511
5,5   1    314   3354 185473 ...
...
T(3,2,2) = 3 because there are 3 distinct dissections of a 3 X 2 X 2 rectangular cuboid into integer-sided cubes. The dissections expanded into 2 dimensions are:
  ._____.    ._____.    ._____.
  |_|_|_|    |_|_|_|    |_|_|_|
  |_|_|_|    |_|_|_|    |_|_|_|
  ._____.    ._____.    ._____.
  |   |_|    |   |_|    |   |_|
  |___|_|    |___|_|    |___|_|
  ._____.    ._____.    ._____.
  |_|   |    |_|   |    |_|   |
  |_|___|    |_|___|    |_|___|

Christopher Hunt Gribble, C++ program

Cf. A219924.

T(1,1,r) = T(n,n,1) = 1. - R. J. Mathar, Dec 03 2017
T(2,2,r) = A000045(r+1). - R. J. Mathar, Dec 03 2017
T(3,3,r>=1) = 1, 5, 10, 31, ... with g.f. 1/(1-x-4*x^2-x^3). - R. J. Mathar, Dec 03 2017
T(4,4,r>=1) = 1, 35, 167, 2098, 15511, 151378, 1272179, 11574563, 100928230, 900224006, ... with TBD rational g.f. - R. J. Mathar, Dec 03 2017
T(n,n,2) = A063443(n). - R. J. Mathar, Dec 03 2017

20 more terms from R. J. Mathar, Dec 03 2017

A334617 a(n) is the number of ways to tile a size n staircase polyomino with staircase polyominoes.

Original entry on oeis.org

1, 2, 8, 57, 806, 20840, 1038266, 97115638, 17213517207, 5768580741287

Offset: 1

Peter Kagey, Sep 08 2020

A size-n staircase polynomo is a polyomino consisting of n left-aligned rows in increasing length of 1, 2, ..., n. Rotations of staircase polyominoes are also polyominoes.

			For n = 3 the a(3) = 8 tilings are:
+---+          +---+          +---+          +---+
|   |          |   |          |   |          |   |
+---+---+      +   +---+      +---+---+      +---+---+
|   |   |      |       |      |   |   |      |   |   |
+---+---+---+, +---+---+---+, +   +---+---+, +---+   +---+,
|   |   |   |  |   |   |   |  |       |   |  |   |       |
+---+---+---+  +---+---+---+  +---+---+---+  +---+---+---+
+---+          +---+          +---+          +---+
|   |          |   |          |   |          |   |
+---+---+      +---+---+      +---+---+      +   +---+
|       |      |       |      |   |   |      |       |
+---+   +---+, +   +---+---+, +---+   +---+, +       +---+.
|   |   |   |  |   |   |   |  |       |   |  |           |
+---+---+---+  +---+---+---+  +---+---+---+  +---+---+---+

Code Golf Stack Exchange user "Bubbler", Tiling a staircase with staircases.

Cf. A000105, A045846, A219924, A254414, A335547.

a(8) from Seiichi Manyama, Sep 09 2020

a(9)-a(10) from Bert Dobbelaere, Sep 12 2020

cp's OEIS Frontend

A224861 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 2 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A224867 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 4 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A160911 a(n) is the number of arrangements of n square tiles with coprime sides in a rectangular frame, counting reflected, rotated or rearranged tilings only once.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A228267 Number T(n,k,r) of dissections of an n X k X r rectangular cuboid into integer-sided cubes including rotations and reflections; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A334617 a(n) is the number of ways to tile a size n staircase polyomino with staircase polyominoes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions