A360630 -id:A360630 - 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

A361217 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X n square.

Original entry on oeis.org

1, 4, 56, 5752, 2519124, 6126859968

Offset: 1

Pontus von Brömssen, Mar 05 2023

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

Main diagonal of A361216.

Cf. A360630, A361222.

A361222 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X n square, up to rotations and reflections.

Original entry on oeis.org

1, 1, 8, 719, 315107

Offset: 1

Pontus von Brömssen, Mar 05 2023

Main diagonal of A361221.

Cf. A361217, A360630.

A361524 Number of ways of dividing an n X n square into integer-sided rectangles, up to rotations and reflections.

Original entry on oeis.org

1, 1, 4, 54, 9235, 10538496, 66906507915, 2262572656817797, 406359897582963166777, 387240433077951047222490766, 1957233446631303872408683778546809, 52459774417987065589052845904624173777442, 7455958280198359250316552005822713102696893557376

Offset: 0

Pontus von Brömssen, Mar 15 2023

Nathan Jellis, Program to calculate up to a(12)

Main diagonal of A361523.

Cf. A182275 (rotations and reflections are considered distinct), A224239 (square pieces), A360630.

Python
```
# See Jellis link.
```

a(n) >= A182275(n)/8.

a(n) ~ A182275(n)/8.

a(6)-a(12) from Nathan Jellis, Aug 25 2025

A361002 Number of tilings of an n X n square by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the square (except rotations and reflections of the original tiling).

Original entry on oeis.org

1, 4, 9, 23, 41

Offset: 1

Pontus von Brömssen, Feb 28 2023

Main diagonal of A361001.

Cf. A360630.

A361425 Maximum difficulty level (see A361424 for the definition) for tiling an n X n square with a set of integer-sided rectangles, rounded down to the nearest integer.

Original entry on oeis.org

1, 2, 8, 80, 1152, 53760

Offset: 1

Pontus von Brömssen, Mar 11 2023

For all currently known terms, the maximum difficulty level is an integer.

Main diagonal of A361424.

Cf. A360630, A361217, A361222.

A363381 a(n) is the number of distinct n-cell patterns that tile an n X n square.

Original entry on oeis.org

1, 2, 1, 60, 1, 102, 1, 62714

Offset: 1

Thomas Young, May 30 2023

Consider n unit squares contained within an n X n square. The n unit squares are an n-cell pattern of the n X n square if, when copied n-1 times, they can, with various rigid transformations, be combined to tessellate the n X n square.
Put another way:
Consider, for example, for n = 4, a transparency with an outline of a 4 X 4 square filled by 16 unit squares. Any 4 unit squares are painted the same color. Those four squares are a potential n-cell pattern of the 4 X 4 square. Three copies of the transparency are made with only the color of the 4 squares being different. If a person can stack the transparencies in such a way that they fill the 4 X 4 square, then the n-cell pattern is acceptable.
Put another way:
n unit squares from an n X n square are painted a color. Those n unit squares are an n-cell pattern. If n-1 copies of the pattern can be painted (each a different color) and together they fill the n X n square, then the n unit squares form an acceptable n-cell pattern.
Conjecture by Andrew Young: For an n X n square, where n is an odd prime, there is only one n-cell pattern.
Conjecture by Andrew Young and Thomas Young: An odd integer n>=3 is prime iff there exists only one n-cell pattern for an n X n square.
For any number n, there is always the 1 X n pattern that tiles the n X n square.
For composite numbers n = f1*f2, 1 < f1 <= f2 < n, there is always an additional f1 X f2 pattern. For example, a 14 X 14 square can be tiled using fourteen 1 X 14 rectangles or fourteen 2 X 7 rectangles; a 15 X 15 square can be tiled using fifteen 1 X 15 rectangles or fifteen 3 X 5 rectangles; a 9 X 9 square can be tiled using nine 1 X 9 rectangles or nine 3 X 3 squares (as in Sudoku!).
The second conjecture is a Corollary of the first: If n = p*q is not prime, then there is always a second tiling using rectangles, as explained above. Since the second conjecture implies the first, the two conjectures are actually equivalent. - M. F. Hasler, Jun 15 2025

			For n = 1, there is one 1-cell pattern because there is only one unit square to paint.
For n = 2, there are two 2-cell patterns:
   +---+---+     +---+---+         +---+
   | 1 | 2 |     | 1 | 2 |         | 1 |
   +---+---+     +---+---+   and   +---+---+
   | 3 | 4 |                           | 4 |
   +---+---+                           +---+
For n = 3, there is one 3-cell pattern:
   +---+---+---+
   | 1 | 2 | 3 |
   +---+---+---+
   | 4 | 5 | 6 |     It is   +---+---+---+
   +---+---+---+             | 1 | 2 | 3 |
   | 7 | 8 | 9 |             +---+---+---+
   +---+---+---+
For n = 4, there are sixty 4-cell patterns:
   +---+---+---+---+
   | 1 | 2 | 3 | 4 |   One is  +---+---+---+---+
   +---+---+---+---+           | 1 | 2 | 3 | 4 |
   | 5 | 6 | 7 | 8 |           +---+---+---+---+
   +---+---+---+---+
   | 9 |10 |11 |12 |     which is equivalent to:
   +---+---+---+---+                       +---+
   |13 |14 |15 |16 |                       | 1 |
   +---+---+---+---+                       +---+
                                           | 5 |
                                           +---+
and therefore these two are not            | 9 |
counted as distinct patterns.              +---+
                                           |13 |
                                           +---+
Another 4-cell pattern for a 4 X 4 square
   +---+---+---+---+
   | x | x | y | y |
   +---+---+---+---+   is
   | z | y | x | a |          +---+---+
   +---+---+---+---+          | x | x |
   | y | z | a | x |          +---+---+---+
   +---+---+---+---+                  | x |
   | a | a | z | z |                  +---+---+
   +---+---+---+---+                      | x |
                                          +---+
     +---+---+
     | x | x |
     +---+---+---+       is equivalent to
             | x |
             +---+---+
                 | x |
                 +---+
           +---+---+  +---+                          +---+
           | y | y |  | z |                          | a |
       +---+---+---+  +---+---+                  +---+---+
       | y |              | z |                  | a |
   +---+---+              +---+---+---+  +---+---+---+
   | y |                      | z | z |  | a | a |
   +---+                      +---+---+  +---+---+
because the shapes can be created through reflection, rotation, or translation.
Therefore, they are counted as one pattern.
For n = 5, there is one 5-cell pattern.

Thomas Young, The 60 4-cell patterns for a 4 X 4 square.
Thomas Young, A Java program to calculate the number of 4-cell pattern for a 4 X 4 square.
Thomas Young, The 102 6-cell patterns for a 6 X 6 square.

Cf. A291806, A227004, A291808, A360630, A291807, A328020, A220778.

a(n) >= 2 if n is composite.

For n > 1, a(n) = 1 iff n is an odd prime (conjectured: cf comments).

a(7)-a(8) from Andrew Howroyd, Jun 04 2023

Minor edits by M. F. Hasler, Jun 15 2025

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.

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A361217 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X n square.

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A361222 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X n square, up to rotations and reflections.

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A361524 Number of ways of dividing an n X n square into integer-sided rectangles, up to rotations and reflections.

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A361002 Number of tilings of an n X n square by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the square (except rotations and reflections of the original tiling).

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A361425 Maximum difficulty level (see A361424 for the definition) for tiling an n X n square with a set of integer-sided rectangles, rounded down to the nearest integer.

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A363381 a(n) is the number of distinct n-cell patterns that tile an n X n square.

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