A226979 -id:A226979 - cp's OEIS frontend

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.

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

A227004 Irregular triangle read by rows: T(n,k) is the number of inequivalent tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 1, 3, 4, 2, 2, 0, 0, 0, 0, 1, 1, 3, 13, 20, 17, 6, 10, 5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 6, 37, 138, 280, 300, 255, 218, 98, 43, 55, 28, 20, 11, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Christopher Hunt Gribble, Jun 26 2013

The n-th row contains (n-1)^2 + 1 elements.
The irregular triangle is shown below.
\ k 0     1     2     3     4     5     6     7     8     9  ...
n
1   1
2   1     1
3   1     1     0     0     1
4   1     3     4     2     2     0     0     0     0     1
5   1     3    13    20    17     6    10     5     0     1  ...
6   1     6    37   138   280   300   255   218    98    43  ...
7   1     6    75   505  2160  5410  8508  9179  8805  7917  ...

			For n = 4, there are 3 inequivalent tilings that contain 1 isolated node, so T(4,1) = 3.
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.  Then the 3 tilings are:
1 1 1 1 1    1 1 1 1 1    1 1 1 1 1
1 0 1 1 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 1 1 1    1 1 1 1 1    1 1 1 1 1
1 1 1 1 1    1 1 1 1 1    1 1 1 1 1

Christopher Hunt Gribble, Table of n, a(n) for n = 1..98
Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004

Cf. A224239.

Sum_{k=0..(n-1)^2} T(n,k) = A224239(n).

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

A240120 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has reflective symmetry in both diagonals and no other reflective symmetries.

Original entry on oeis.org

0, 0, 0, 1, 1, 9, 19, 121, 275, 2489, 7217, 86775

Offset: 1

Ed Wynn, Apr 01 2014

'Inequivalent' has the same sense as in A224239: we do not regard dissections that differ by a rotation and/or reflection as distinct.

The two reflective symmetries imply 180-degree (but not 90-degree) rotational symmetry.

			This is the single dissection for n=4:
---------
|   | | |
|   -----
|   | | |
---------
| | |   |
-----   |
| | |   |
---------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226979, A045846, A224239, A240121, A240122.

A240121 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has two reflective symmetries in axes parallel to the sides, and no other reflective symmetries.

Original entry on oeis.org

0, 0, 0, 1, 0, 13, 5, 183, 75, 4408, 1501, 180324

Offset: 1

Ed Wynn, Apr 01 2014

The two reflective symmetries imply 180-degree (but not 90-degree) rotational symmetry.

			This dissection is the only example for n=4:
---------
| |   | |
---   ---
| |   | |
---------
| |   | |
---   ---
| |   | |
---------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226979, A045846, A224239, A240120, A240122.

A240122 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 90-degree rotational symmetry and no reflective symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 12, 40, 154, 760, 3260, 22730

Offset: 1

Ed Wynn, Apr 01 2014

			The two dissections for n=6:
-------------    -------------
| |   | | | |    | |   | | | |
---   -------    ---   -------
| |   | |   |    | |   | |   |
---------   |    ---------   |
| | |   |   |    | | | | |   |
-----   -----    -------------
|   |   | | |    |   | | | | |
|   ---------        ---------
|   | |   | |    |   | |   | |
-------   ---    -------   ---
| | | |   | |    | | | |   | |
-------------    -------------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226979, A045846, A224239, A240120, A240121.

cp's OEIS Frontend

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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A227004 Irregular triangle read by rows: T(n,k) is the number of inequivalent tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

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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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A240120 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has reflective symmetry in both diagonals and no other reflective symmetries.

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A240121 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has two reflective symmetries in axes parallel to the sides, and no other reflective symmetries.

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A240122 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 90-degree rotational symmetry and no reflective symmetry.

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