A219861 -id:A219861 - cp's OEIS frontend

A108066 Number of distinct ways to dissect a square into n rectangles of equal area.

1, 1, 2, 6, 18, 65, 281, 1343, 6953, 38023

Offset: 1

Hans Riesebos (hans.riesebos(AT)wanadoo.nl) and Herman Beeksma, Jun 03 2005

"Distinct" here means that dissections differing only by a rotation and/or reflection are not counted as different (see A189243).

The first time the pieces can be made to all have different shapes (but the same area) is at n=7 - see Descartes (1971) and the illustration; also Wells, Weisstein. - N. J. A. Sloane, Dec 05 2012

			There are six ways to dissect a square into four rectangles of equal area, so a(4)=6:
+-+-----+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+
| |     | | | |   | | |     | | | | | | | |   | | |   |   |
| |     | | | |   | | |     | | | | | | | |   | | |   |   |
| +--+--+ | | |   | | +-----+ | | | | | | |   | | |   |   |
| |  |  | | | +---+ | |     | | | | | | | +---+ | +---+---+
| |  |  | | | |   | | |_____| | | | | | | |   | | |   |   |
| |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
| |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
+-+--+--+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+

David Wells, Penguin Dictionary of Curious and Interesting Geometry, 1991, pp. 15-16.

Blanche Descartes, Divisions of a square into rectangles, Eureka, No. 34 (1971), 31-35.
R. Häggkvist, P.-O. Lindberg, and B. Lindström, Dissecting a square into rectangles of equal area, Discr. Math. 47 (1983), 321-323.
Johan Nilsson, Illustrations for terms up to a(6)..
N. J. A. Sloane, Blanche Descartes's dissection into 7 pieces with equal areas but different shapes.
Eric Weisstein, Blanche's Dissection.

Cf. A189243, A219861, A359146.

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

Original entry on oeis.org

1, 2, 6, 21, 88, 390, 1914

Offset: 1

Yi Yang, Apr 19 2011

Dissections which differ by rotations or reflections are counted as distinct.
Rectangles may have different shapes.
a(1) to a(5) are the same (but not a(6)) as:
  A033540 a(n+1) = n*(a(n)+1), n >= 1, a(1) = 1.
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

			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)

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

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

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

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

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A108066 Number of distinct ways to dissect a square into n rectangles of equal area.

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