A210517 - OEIS

%I #35 Jan 12 2023 01:29:57

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

%N Number of rectangles dissectable into n squares, unique up to aspect ratio.

%C The rectangles are distinguishable by aspect ratio, not size.

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

%H Rainer Rosenthal, <a href="/A210517/a210517.txt">SetA210517(n) = set of aspect ratios for n squares (for n < 9).</a>

%e 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.

%e 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.

%e 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.

%e 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.

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

%K nonn,hard,more

%O 1,3

%A _Geoffrey H. Morley_, Jan 26 2013

%E Title changed by _Rainer Rosenthal_, Dec 30 2022

%E a(7) corrected, a(8) new. - _Marx Stampfli_ and _Rainer Rosenthal_, Jan 10 2023