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A359146 Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible. 2
1, 1, 3, 11, 51, 245, 1372 (list; graph; refs; listen; history; text; internal format)



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

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?


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


Table of n, a(n) for n=1..7.

John Baez, Rahul Narain, Daniel Piker, Lisanne Taams, Ian Henderson, David Eppstein, and others, Dividing a Square into Similar Rectangles, First posted Dec. 22 2022, with many later comments

John Baez, Dividing a Square into 7 Similar Rectangles, Mar 06 2023. Continuation of first blog entry, with a new value for a(7).

Ian Henderson, Cutting squares into similar rectangles using a computer program, Feb 02 2023

Siobhan Roberts, The Quest to Find Rectangles in a Square, New York Times, Feb 07 2023

N. J. A. Sloane, Illustration for a(3) = 3


Cf. A100664, A108066, A348456.

Sequence in context: A145144 A284702 A191341 * A199212 A113283 A326519

Adjacent sequences: A359143 A359144 A359145 * A359147 A359148 A359149




N. J. A. Sloane, Feb 07 2023


It appears that a(8) = 8522. - Ian Henderson, Feb 07, 2023, corrected Mar 07 2023

a(7) corrected by N. J. A. Sloane, Mar 07 2023, based on the second John Baez blog entry.



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Last modified March 25 19:21 EDT 2023. Contains 361528 sequences. (Running on oeis4.)