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A189243 Number of ways to dissect a nonsquare rectangle into n rectangles with equal area.

%I A189243 #40 May 01 2013 21:06:47
%S A189243 1,2,6,21,88,390,1914
%N A189243 Number of ways to dissect a nonsquare rectangle into n rectangles with equal area.
%C A189243 Dissections which differ by rotations or reflections are counted as distinct.
%C A189243 Rectangles may have different shapes.
%C A189243 a(1) to a(5) are the same (but not a(6)) as:
%C A189243   A033540 a(n+1) = n*(a(n)+1), n >= 1, a(1) = 1.
%C A189243 If the dissections with a cross (where four squares share a vertex) were counted twice then a(1) to a(5) would be the same as the 'guillotine partitions' counted by A006318. - _Geoffrey H. Morley_, Dec 31 2012
%H A189243 Geoffrey H. Morley, <a href="/A189243/a189243_1.pdf">Illustration of terms up to a(5)</a>
%H A189243 N. J. A. Sloane, <a href="/A189243/a189243.jpg">Illustration of the term a(4) = 21</a>
%H A189243 "056254628", <a href="http://bbs.emath.ac.cn/thread-2916-1-4.html">A Chinese web page containing the problem and illustrating the initial terms</a>
%F A189243 For n > 4, a(n) = b(n)+
%F A189243 +-------+   +-------+   +-------+   +---+---+   +---+---+
%F A189243 |       |   |       |   |       |   |   |   |   |   |   |
%F A189243 +-------+   +-------+   +-------+   +---+---+   +---+---+
%F A189243 |[a(n-1)|   |       |   |       |   |[a(n-2)|   |       |
%F A189243 |-a(n-2)|*4+| a(n-2)|*2+| a(n-3)|*4+|-a(n-3)|*4+| a(n-4)|*2
%F A189243 |-a(n-3)|   +-------+   +---+---+   |-a(n-4)|   +---+---+
%F A189243 |]      |   |       |   |   |   |   |]      |   |   |   |
%F A189243 +-------+   +-------+   +---+---+   +-------+   +---+---+
%F A189243 = b(n)+4*a(n-1)+2*a(n-2)-4*a(n-3)-2*a(n-4) where b(n) is the number of tilings in which no side of the rectangle comprises the side of a tile or the equal sides of two congruent tiles. For example, b(5) = 2. '*2' counts, say, rotation clockwise by 90 degrees (and rescaling the aspect ratio), while '*4' counts all rotations. - _Geoffrey H. Morley_, Dec 07 2012
%e A189243 There are 6 ways to form a rectangle from 3 rectangles with same area:
%e A189243 +-----+ +-+-+-+ +-----+ +--+--+ +-+---+ +---+-+
%e A189243 |     | | | | | |     | |  |  | | |   | |   | |
%e A189243 +-----+ | | | | +--+--+ |  |  | | |   | |   | |
%e A189243 |     | | | | | |  |  | |  |  | | +---+ +---+ |
%e A189243 +-----+ | | | | |  |  | +--+--+ | |   | |   | |
%e A189243 |     | | | | | |  |  | |     | | |   | |   | |
%e A189243 +-----+ +-+-+-+ +--+--+ +-----+ +-+---+ +---+-+
%e A189243 So a(3)=6.
%e A189243 From _Geoffrey H. Morley_, Dec 03 2012: (Start)
%e A189243 b(n) in the given formula is the sum of the appropriate tilings from certain 'frames'. A number that appears in a subrectangle in a frame is the number of rectangles into which the subrectangle is to be divided. Tilings are also counted that are from a reflection and/or half-turn of the frame.
%e A189243 For n = 6 there are 3(X2) frames:
%e A189243 +---+-+-+  +-+-----+  +-+-----+
%e A189243 |   | | |  | |     |  | |     |
%e A189243 |   | | |  | +---+-+  | |  2  |
%e A189243 +-+-+ | |  | |   | |  | |     |
%e A189243 | | | | |  | +---+ |  | +---+-+
%e A189243 | | +-+-+  | |   | |  | |   | |
%e A189243 | | |   |  +-+---+ |  +-+---+ |
%e A189243 | | |   |  |     | |  |     | |
%e A189243 +-+-+---+  +-----+-+  +-----+-+
%e A189243   2 ways     2 ways     8 ways
%e A189243 The only other frames which yield desired tilings are obtained by rotating each frame above by 90 degrees and scaling it to fit a rectangle with the inverse aspect ratio.
%e A189243 So b(6) = 2(2+2+8) = 24, and a(6) = b(6)+4*a(5)+2*a(4)-4*a(3)-2*a(2) = 24+4*88+2*21-4*6-2*2 = 390.
%e A189243 For n = 7 we can use 7(X2) frames:
%e A189243 +---+--+
%e A189243 |   |  |
%e A189243 |   |  |
%e A189243 | 4 |3 |
%e A189243 |   |  |
%e A189243 |   |  |
%e A189243 |   |  |
%e A189243 +---+--+
%e A189243 63 ways [of creating tilings counted by b(7)]
%e A189243 +---+--+  +-+----+  +--+---+  +-----++  +--+---+  +----+-+
%e A189243 |   |  |  | |    |  |  |   |  ++----+|  |  |   |  ++-+-+ |
%e A189243 |   +-++  | +---++  |2 | 2 |  ||    ||  |  +-+-+  || | | |
%e A189243 | 3 | ||  |2|   ||  |  +--++  ||    ||  |2 | | |  || | | |
%e A189243 |   | ||  | | 2 ||  |  |  ||  || 3  ||  |  | | |  || +-+-+
%e A189243 |   | ||  | |   ||  +--+--+|  ||    ||  +--+-+2|  || |   |
%e A189243 +---+-+|  +-+---+|  |     ||  |+----++  |    | |  |+-+---+
%e A189243 +-----++  +-----++  +-----++  ++-----+  +----+-+  ++-----+
%e A189243 24 ways   16 ways   12 ways   10 ways    8 ways    4 ways
%e A189243 As for n = 6, these are only half the frames and tilings.
%e A189243 So b(7) = 2(63+24+16+12+10+8+4) = 274, and a(7) = b(7)+4*a(6)+2*a(5)-4*a(4)-2*a(3) = 274+4*390+2*88-4*21-2*6 = 1914.
%e A189243 (End)
%Y A189243 See the analogous sequences A219861 and A108066 where we count dissections up to symmetry of nonsquare rectangles and squares respectively. - _Geoffrey H. Morley_, Dec 03 2012
%K A189243 nonn,nice,more
%O A189243 1,2
%A A189243 _Yi Yang_, Apr 19 2011
%E A189243 Edited by _N. J. A. Sloane_, Apr 21 2011
%E A189243 a(7) added by _Geoffrey H. Morley_, Dec 03 2012
%E A189243 a(7) corrected by _Geoffrey H. Morley_, Dec 05 2012