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

Author

N. J. A. Sloane, Feb 07 2023

Keywords

• nonn
• more

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

• 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.

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.