A224239 -id:A224239 - cp's OEIS frontend

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.

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.

A240123 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has a reflective symmetry in one diagonal, but no other symmetries.

Original entry on oeis.org

0, 0, 1, 3, 19, 107, 847, 8647, 119835, 2255123, 58125783, 2050662011

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 three dissections for n=4:
---------    ---------    ---------
|   | | |    |   |   |    |     | |
|   -----    |   |   |    |     ---
|   | | |    |   |   |    |     | |
---------    ---------    |     ---
| | | | |    |   | | |    |     | |
---------    |   -----    ---------
| | | | |    |   | | |    | | | | |
---------    ---------    ---------

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

Cf. A226980, A045846, A224239, A240124, A240125.

A240124 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 180-degree rotational symmetry, but no other symmetries.

Original entry on oeis.org

0, 0, 0, 0, 2, 19, 109, 1781, 13660, 397689, 5368943, 289864745

Offset: 1

Ed Wynn, Apr 01 2014

			The two dissections for n=5:
-----------    -----------
|   |   | |    | |   | | |
|   |   ---    ---   -----
|   |   | |    | |   | | |
-----------    -----------
| | | | | |    | | | | | |
-----------    -----------
| |   |   |    | | |   | |
---   |   |    -----   ---
| |   |   |    | | |   | |
-----------    -----------

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

Cf. A226980, A045846, A224239, A240123, A240125.

A240125 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has one reflective symmetry in an axis parallel to a side, but no other symmetries.

Original entry on oeis.org

0, 0, 0, 3, 5, 138, 201, 13032, 19990, 4095612, 7026883, 4451051502

Offset: 1

Ed Wynn, Apr 01 2014

			The three dissections for n=4, with the axis horizontal:
---------    ---------    ---------
|   | | |    |   | | |    | | | | |
|   -----    |   -----    ---------
|   | | |    |   |   |    |   | | |
---------    -----   |    |   -----
|   | | |    |   |   |    |   | | |
|   -----    |   -----    ---------
|   | | |    |   | | |    | | | | |
---------    ---------    ---------

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

Cf. A226980, A045846, A224239, A240123, A240124.

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

A358716 a(n) is the number of inequivalent ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

Original entry on oeis.org

1, 2, 3, 12, 50, 711, 18031, 952013, 92323440

Offset: 1

Craig Knecht and John Mason, Nov 28 2022

Similar to A358715, but now we do not regard dissections which differ by a rotation and/or reflection as distinct.

			a(3)=3 because of:
    /\      /\      /\
   /  \    /\/\    /\/\
  /    \  /  \/\  /\/\/\

Cf. A224239, A358715.

Cf. A299705.

A362259 Maximum number of ways in which a set of integer-sided squares can tile an n X n square, up to rotations and reflections.

Original entry on oeis.org

1, 1, 1, 1, 4, 20, 277, 7855, 487662

Offset: 0

Pontus von Brömssen, Apr 15 2023

Main diagonal of A362258.

Cf. A224239, A236679, A361222 (rectangular pieces), A362143.

a(n) >= A362143(n)/8.

cp's OEIS Frontend

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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A240123 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has a reflective symmetry in one diagonal, but no other symmetries.

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A240124 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 180-degree rotational symmetry, but no other symmetries.

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A240125 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has one reflective symmetry in an axis parallel to a side, but no other symmetries.

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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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A358716 a(n) is the number of inequivalent ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

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A362259 Maximum number of ways in which a set of integer-sided squares can tile an n X n square, up to rotations and reflections.

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