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

1,2

Comments

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

Links

Table of n, a(n) for n=1..12.

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

Example

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.

Crossrefs

Main diagonal of A227690.

Cf. A045846, A034295, A219924.

Sequence in context: A061912 A212433 A013167 * A361520 A068096 A098462

Adjacent sequences: A224236 A224237 A224238 * A224240 A224241 A224242

Keyword

nonn,hard,more

Author

N. J. A. Sloane, Apr 15 2013

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

Status

approved