A227004 -id:A227004 - cp's OEIS frontend

A226979 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 2 elements.

0, 0, 0, 2, 2, 24, 36, 344, 504, 7657, 11978, 289829

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

			For n=5, there are 2 dissections where the orbits under the symmetry group of the square, D4, have 2 elements.
For n=4, the 2 dissections can be seen in A240120 and A240121.

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A045846, A034295, A219924, A224239, A226978, A226980, A226981, A240120, A240121, A240122.

A226978(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).
1*A226978(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).
A226979(n) = A240120(n) + A240121(n) + A240122(n).

a(8)-a(12) from Ed Wynn, Apr 01 2014

A226980 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

Original entry on oeis.org

0, 0, 1, 6, 26, 264, 1157, 23460, 153485, 6748424, 70521609, 6791578258

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

			For n=5, there are 26 dissections where the orbits under the symmetry group of the square, D4, have 4 elements.
The 6 dissections for n=4 can be seen in A240123 and A240125.

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A045846, A034295, A219924, A224239, A226978, A226979, A226981, A240123, A240124, A240125.

A226978(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).
1*A226978(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).
A226980(n) = A240123(n) + A240124(n) + A240125(n).

a(8)-a(12) from Ed Wynn, Apr 01 2014

A226978 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element.

Original entry on oeis.org

1, 2, 2, 4, 4, 12, 8, 44, 32, 228, 148, 1632, 912, 16004, 8420, 213680, 101508, 3933380, 1691008, 98949060, 38742844, 3413919788, 1213540776, 161410887252, 52106993880

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

From Walter Trump, Dec 15 2022: (Start)
a(n) is the number of fully symmetric dissections of an n X n square into squares with integer sides.
Conjecture: For n>3 the number of dissections is a multiple of 4. (End)

			For n=5, there are 4 dissections where the orbits under the symmetry group of the square, D4, have 1 element.
For n=4, 3 dissections divide the square into uniform subsquares (of sizes 1, 2 and 4 respectively), and this is the 4th:
---------
| | | | |
---------
| |   | |
---   ---
| |   | |
---------
| | | | |
---------

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Walter Trump, Example for n=19

Cf. A045846, A034295, A219924, A224239, A226979, A226980, A226981.

a(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).

1*a(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).

a(8)-a(12) from Ed Wynn, Apr 02 2014

a(13)-a(25) from Walter Trump, Dec 15 2022

A226981 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 8 elements.

Original entry on oeis.org

0, 0, 0, 1, 45, 1194, 55777, 4471175, 669049507, 187616301623, 98793450008033, 97702667035688951

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

			For n=5, there are 45 dissections where the orbits under the symmetry group of the square, D4, have 8 elements.
For n=4, this is the only dissection:
---------
|   | | |
|   -----
|   |   |
-----   |
| | |   |
---------
| | | | |
---------

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A045846, A034295, A219924, A224239, A226978, A226979, A226980.

A226978(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).

1*A226978(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).

a(8)-a(12) from Ed Wynn, Apr 02 2014

A225803 Number T(n,k,u) of tilings of an n X k rectangle using integer-sided square tiles, reduced for symmetry, containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k < n, u >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 1, 1, 1, 2, 2, 1, 2, 4, 0, 2, 1, 1, 4, 13, 10, 6, 3, 1, 0, 0, 1, 1, 1, 3, 4, 1, 1, 3, 8, 3, 2, 3, 0, 0, 1, 1, 6, 23, 33, 24, 15, 6, 0, 2, 2, 2, 1, 1, 6, 40, 101, 129, 79, 74, 53, 13, 9, 11, 4, 0, 0, 0, 0, 1

Offset: 1

Christopher Hunt Gribble, Jul 28 2013

The number of entries per row is given by A225568(n>0 and n != A000217(1:)).

			The irregular triangle T(n,k,u) begins:
n,k\u  0   1   2   3   4   5   6   7   8   9  10  11  12 ...
2,1    1
3,1    1
3,2    1   1
4,1    1
4,2    1   2   1
4,3    1   2   2   0   1
5,1    1
5,2    1   2   2
5,3    1   2   4   0   2   1
5,4    1   4  13  10   6   3   1   0   0   1
6,1    1
6,2    1   3   4   1
6,3    1   3   8   3   2   3   0   0   1
6,4    1   6  23  33  24  15   6   0   2   2   1
6,5    1   6  40 101  79  74  53  13   9  11   4   0   0 ...
...
T(5,3,2) = 4 because there are 4 different sets of tilings of the 5 X 3 rectangle by integer-sided squares in which each tiling contains 2 isolated nodes.  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
A 2 X 2 square contains 1 isolated node.  Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  An example of a tiling in each set is:
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 0 1 1    1 0 1 1    1 0 1 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 1 0 1    1 1 1 1    1 1 1 1
   1 1 1 1    1 1 1 1    1 0 1 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1

Christopher Hunt Gribble, Rows 1..28 for n = 2..8 and k = 1..n-1 flattened
Christopher Hunt Gribble, C++ program

Cf. A224239, A227004, A227690, A224850, A224861, A224867, A225777, A225542.

T1(n,k,0) = 1, T1(n,k,1) = floor(n/2)*floor(k/2).

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

A226979 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 2 elements.

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A226980 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

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A226978 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element.

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A226981 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 8 elements.

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A225803 Number T(n,k,u) of tilings of an n X k rectangle using integer-sided square tiles, reduced for symmetry, containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k < n, u >= 0, read by rows.

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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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