A240122 -id:A240122 - 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

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.

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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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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