A189243 Number of ways to dissect a nonsquare rectangle into n rectangles with equal area.
1, 2, 6, 21, 88, 390, 1914
Offset: 1
Examples
There are 6 ways to form a rectangle from 3 rectangles with same area: +-----+ +-+-+-+ +-----+ +--+--+ +-+---+ +---+-+ | | | | | | | | | | | | | | | | | +-----+ | | | | +--+--+ | | | | | | | | | | | | | | | | | | | | | | +---+ +---+ | +-----+ | | | | | | | +--+--+ | | | | | | | | | | | | | | | | | | | | | | | +-----+ +-+-+-+ +--+--+ +-----+ +-+---+ +---+-+ So a(3)=6. From _Geoffrey H. Morley_, Dec 03 2012: (Start) 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. For n = 6 there are 3(X2) frames: +---+-+-+ +-+-----+ +-+-----+ | | | | | | | | | | | | | | | +---+-+ | | 2 | +-+-+ | | | | | | | | | | | | | | | +---+ | | +---+-+ | | +-+-+ | | | | | | | | | | | | +-+---+ | +-+---+ | | | | | | | | | | | +-+-+---+ +-----+-+ +-----+-+ 2 ways 2 ways 8 ways 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. 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. For n = 7 we can use 7(X2) frames: +---+--+ | | | | | | | 4 |3 | | | | | | | | | | +---+--+ 63 ways [of creating tilings counted by b(7)] +---+--+ +-+----+ +--+---+ +-----++ +--+---+ +----+-+ | | | | | | | | | ++----+| | | | ++-+-+ | | +-++ | +---++ |2 | 2 | || || | +-+-+ || | | | | 3 | || |2| || | +--++ || || |2 | | | || | | | | | || | | 2 || | | || || 3 || | | | | || +-+-+ | | || | | || +--+--+| || || +--+-+2| || | | +---+-+| +-+---+| | || |+----++ | | | |+-+---+ +-----++ +-----++ +-----++ ++-----+ +----+-+ ++-----+ 24 ways 16 ways 12 ways 10 ways 8 ways 4 ways As for n = 6, these are only half the frames and tilings. 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. (End)
Links
- Geoffrey H. Morley, Illustration of terms up to a(5)
- N. J. A. Sloane, Illustration of the term a(4) = 21
- "056254628", A Chinese web page containing the problem and illustrating the initial terms
Crossrefs
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
Formula
For n > 4, a(n) = b(n)+
+-------+ +-------+ +-------+ +---+---+ +---+---+
| | | | | | | | | | | |
+-------+ +-------+ +-------+ +---+---+ +---+---+
|[a(n-1)| | | | | |[a(n-2)| | |
|-a(n-2)|*4+| a(n-2)|*2+| a(n-3)|*4+|-a(n-3)|*4+| a(n-4)|*2
|-a(n-3)| +-------+ +---+---+ |-a(n-4)| +---+---+
|] | | | | | | |] | | | |
+-------+ +-------+ +---+---+ +-------+ +---+---+
= 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
Extensions
Edited by N. J. A. Sloane, Apr 21 2011
a(7) added by Geoffrey H. Morley, Dec 03 2012
a(7) corrected by Geoffrey H. Morley, Dec 05 2012
Comments