cp's OEIS Frontend

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

%I A224239 #71 Mar 18 2023 08:37:26
%S A224239 1,2,3,13,77,1494,56978,4495023,669203528,187623057932,98793520541768,
%T A224239 97702673827558670
%N A224239 Number of inequivalent ways to cut an n X n square into squares with integer sides.
%C A224239 Similar to A045846, but now we do not regard dissections which differ by a rotation and/or reflection as distinct.
%H A224239 Don Reble, <a href="/A224239/a224239_1.txt">C programs for A224239</a>
%H A224239 Don Reble, <a href="/A224239/a224239.txt">Comments on the calculation of a(10)</a>
%H A224239 N. J. A. Sloane, <a href="/A224239/a224239_4.jpg">Illustration of the first five terms, page 1 of 4</a> (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).)
%H A224239 N. J. A. Sloane, <a href="/A224239/a224239_5.jpg">Illustration of the first five terms, page 2 of 4</a> (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
%H A224239 N. J. A. Sloane, <a href="/A224239/a224239_6.jpg">Illustration of the first five terms, page 3 of 4</a> (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
%H A224239 N. J. A. Sloane, <a href="/A224239/a224239_7.jpg">Illustration of the first five terms, page 4 of 4</a> (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
%H A224239 Ed Wynn, <a href="http://arxiv.org/abs/1308.5420">Exhaustive generation of Mrs Perkins's quilt square dissections for low orders</a>, 2013; arXiv:1308.5420
%e A224239 For n=5, the illustrations (see links) show that the 77 solutions consist of:
%e A224239 4 dissections each with 1 image under the group of the square, for a total of 4,
%e A224239 2 dissections each with 2 images under the group of the square, totaling 4,
%e A224239 26 dissections each with 4 images under the group of the square, totaling 104, and
%e A224239 45 dissections each with 8 images under the group of the square, totaling 360,
%e A224239 for a grand total of 77 dissections with 472 images, agreeing with A045846(5) = 472.
%Y A224239 Main diagonal of A227690.
%Y A224239 Cf. A045846, A034295, A219924.
%K A224239 nonn,hard,more
%O A224239 1,2
%A A224239 _N. J. A. Sloane_, Apr 15 2013
%E A224239 a(6)-a(10) from _Don Reble_, Apr 15 2013
%E A224239 a(11)-a(12) from Ed Wynn, 2013. - _N. J. A. Sloane_, Nov 29 2013