A210517 Number of rectangles dissectible into n squares, unique up to aspect ratio.

1, 1, 2, 5, 11, 28, 74, 211

Offset: 1

Author

Geoffrey H. Morley, Jan 26 2013

Keywords

• nonn
• hard
• more

Comments

The rectangles are distinguishable by aspect ratio, not size.

A rectangle is dissectible into squares if and only if its sides are commensurable. A rectangle with commensurable sides is dissectible into n squares for all but a finite number of positive integers n. For example, a square is dissectible into any number of squares other than 2, 3, or 5.

Examples

			For n = 3 the a(3) = 2 rectangles are 3 X 1 and 3 X 2 with aspect ratio 3/1 and 3/2. For example, a 3 X 2 rectangle can be tiled by a 2 X 2 square and two 1 X 1 squares.
For n = 4 the a(4) = 5 aspect ratios are 1/1, 4/1, 4/3, 5/2 and 5/3. Ratio 1/1 stems from the square 2 X 2, tiled by four 1 X 1 squares.
For n = 5 the a(5) = 11 aspect ratios are 2/1, 5/1, 5/4, 6/5, 7/2, 7/3, 7/4, 7/5, 7/6, 8/3 and 8/5.
For n = 6 the a(6) = 28 aspect ratios are 1/1, 3/1, 3/2, 4/3, 5/4, 6/1, 6/5, 9/2, 9/4, 9/5, 9/7, 10/3, 10/7, 10/9, 11/3, 11/4, 11/5, 11/6, 11/7, 11/8, 11/10, 12/5, 12/7, 13/5, 13/6, 13/7, 13/8 and 13/11.
		

Links

• Rainer Rosenthal, SetA210517(n) = set of aspect ratios for n squares (for n < 9).

Crossrefs

Cf. A160911 (tilings with same aspect ratio allowed), A221839.

Extensions

Title changed by Rainer Rosenthal, Dec 30 2022

a(7) corrected, a(8) new. - Marx Stampfli and Rainer Rosenthal, Jan 10 2023