A359146 Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible.

1, 1, 3, 11, 51, 245, 1372

Offset

1,3

Comments

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?

References

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

Links

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

Crossrefs

Cf. A100664, A108066, A348456.

Sequence in context: A367560 A284702 A191341 * A199212 A113283 A326519

Adjacent sequences: A359143 A359144 A359145 * A359147 A359148 A359149

Keyword

nonn,more

Author

N. J. A. Sloane, Feb 07 2023

Extensions

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.

Status

approved