A359146 - OEIS

%I #78 Mar 07 2023 10:20:35

%S 1,1,3,11,51,245,1372

%N Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible.

%C Only the proportions of the rectangles are counted, not how the rectangles are arranged in the square.

%C The number b(n) of different ways to divide a square into n similar rectangles, up to rotation and reflection, is at least a(n). We have b(1)=1, b(2)=1, b(3)=3, and Baez remarks that b(4) > 11. It would be nice to know more. Is the sequence {b(n)} already in the OEIS?

%D Siobhan Roberts, An Online Puzzle Excites Math Fans, New York Times, Feb 07 2023, pages D1 and D4.

%H John Baez, Rahul Narain, Daniel Piker, Lisanne Taams, Ian Henderson, David Eppstein, and others, <a href="https://golem.ph.utexas.edu/category/2022/12/dividing_a_square_into_similar.html">Dividing a Square into Similar Rectangles</a>, First posted Dec. 22 2022, with many later comments

%H John Baez, <a href="https://johncarlosbaez.wordpress.com/2023/03/06/dividing-a-square-into-7-similar-rectangles/">Dividing a Square into 7 Similar Rectangles</a>, Mar 06 2023. Continuation of first blog entry, with a new value for a(7).

%H Ian Henderson, <a href="http://ianhenderson.org/similar-rectangles/">Cutting squares into similar rectangles using a computer program</a>, Feb 02 2023

%H Siobhan Roberts, <a href="https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html">The Quest to Find Rectangles in a Square</a>, New York Times, Feb 07 2023

%H N. J. A. Sloane, <a href="/A359146/a359146.pdf">Illustration for a(3) = 3</a>

%Y Cf. A100664, A108066, A348456.

%K nonn,more

%O 1,3

%A _N. J. A. Sloane_, Feb 07 2023

%E It appears that a(8) = 8522. - _Ian Henderson_, Feb 07, 2023, corrected Mar 07 2023

%E a(7) corrected by _N. J. A. Sloane_, Mar 07 2023, based on the second John Baez blog entry.