A224239 Number of inequivalent ways to cut an n X n square into squares with integer sides.

1, 2, 3, 13, 77, 1494, 56978, 4495023, 669203528, 187623057932, 98793520541768, 97702673827558670

Offset: 1

Author

N. J. A. Sloane, Apr 15 2013

Keywords

• nonn
• hard
• more

Comments

Similar to A045846, but now we do not regard dissections which differ by a rotation and/or reflection as distinct.

Examples

			For n=5, the illustrations (see links) show that the 77 solutions consist of:
4 dissections each with 1 image under the group of the square, for a total of 4,
2 dissections each with 2 images under the group of the square, totaling 4,
26 dissections each with 4 images under the group of the square, totaling 104, and
45 dissections each with 8 images under the group of the square, totaling 360,
for a grand total of 77 dissections with 472 images, agreeing with A045846(5) = 472.
		

Links

• Don Reble, C programs for A224239
• Don Reble, Comments on the calculation of a(10)
• N. J. A. Sloane, Illustration of the first five terms, page 1 of 4 (Each dissection is labeled with the number of its images under the symmetry group of the square. The sum of these numbers is A045846(n).)
• N. J. A. Sloane, Illustration of the first five terms, page 2 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
• N. J. A. Sloane, Illustration of the first five terms, page 3 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
• N. J. A. Sloane, Illustration of the first five terms, page 4 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
• Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, 2013; arXiv:1308.5420

Crossrefs

Main diagonal of A227690.

Cf. A045846, A034295, A219924.

Extensions

a(6)-a(10) from Don Reble, Apr 15 2013

a(11)-a(12) from Ed Wynn, 2013. - N. J. A. Sloane, Nov 29 2013