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

%I A210517 #39 Apr 30 2025 14:45:24
%S A210517 1,1,2,5,11,28,74,211
%N A210517 Number of rectangles dissectible into n squares, unique up to aspect ratio.
%C A210517 The rectangles are distinguishable by aspect ratio, not size.
%C A210517 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.
%H A210517 Rainer Rosenthal, <a href="/A210517/a210517.txt">SetA210517(n) = set of aspect ratios for n squares (for n < 9).</a>
%e A210517 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 A210517 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 A210517 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 A210517 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 A210517 Cf. A160911 (tilings with same aspect ratio allowed), A221839.
%K A210517 nonn,hard,more
%O A210517 1,3
%A A210517 _Geoffrey H. Morley_, Jan 26 2013
%E A210517 Title changed by _Rainer Rosenthal_, Dec 30 2022
%E A210517 a(7) corrected, a(8) new. - _Marx Stampfli_ and _Rainer Rosenthal_, Jan 10 2023