A108066 -id:A108066 - cp's OEIS frontend

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

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

A100664 Number of inequivalent ways to dissect a square into n rectangles of equal perimeter.

Original entry on oeis.org

1, 1, 2, 6, 16, 50, 177, 664, 2532, 9785

Offset: 1

Herman Beeksma and Hans Riesebos (hans.riesebos(AT)wanadoo.nl), Aug 04 2005

Dissections differing only by rotations and/or reflections are not counted as distinct. - N. J. A. Sloane, Dec 06 2012

			There are six ways to dissect a square into four rectangles of equal perimeter, so a(4)=6.

Johan Nilsson, Illustrations of terms up to a(6).

Cf. A108066.

A219861 Number of ways to dissect a nonsquare rectangle into n rectangles of equal area up to symmetry.

Original entry on oeis.org

1, 2, 4, 11, 35, 130, 562, 2685, 13901, 76046

Offset: 1

Geoffrey H. Morley, Nov 29 2012

			There are 4 ways (up to symmetry) to form a nonsquare rectangle from 3 rectangles with the same area:
+-----+ +-+-+-+ +-----+ +-+---+
|     | | | | | |     | | |   |
+-----+ | | | | +--+--+ | |   |
|     | | | | | |  |  | | +---+
+-----+ | | | | |  |  | | |   |
|     | | | | | |  |  | | |   |
+-----+ +-+-+-+ +--+--+ +-+---+
So a(3)=4.
The eleven solutions for n=4 can be seen as a subset of the illustration of A189243(4) = 21 in that entry. - _N. J. A. Sloane_, Dec 05 2012

Cf. A108066, A189243.

a(7)-a(10) from Geoffrey H. Morley, Dec 16 2012

A359146 Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible.

Original entry on oeis.org

1, 1, 3, 11, 51, 245, 1372

Offset: 1

N. J. A. Sloane, Feb 07 2023

Only the proportions of the rectangles are counted, not how the rectangles are arranged in the square.

The number b(n) of different ways to divide a square into n similar rectangles, up to rotation and reflection, is at least a(n). We have b(1)=1, b(2)=1, b(3)=3, and Baez remarks that b(4) > 11. It would be nice to know more. Is the sequence {b(n)} already in the OEIS?

Siobhan Roberts, An Online Puzzle Excites Math Fans, New York Times, Feb 07 2023, pages D1 and D4.

John Baez, Rahul Narain, Daniel Piker, Lisanne Taams, Ian Henderson, David Eppstein, and others, Dividing a Square into Similar Rectangles, First posted Dec. 22 2022, with many later comments
John Baez, Dividing a Square into 7 Similar Rectangles, Mar 06 2023. Continuation of first blog entry, with a new value for a(7).
Ian Henderson, Cutting squares into similar rectangles using a computer program, Feb 02 2023
Siobhan Roberts, The Quest to Find Rectangles in a Square, New York Times, Feb 07 2023
N. J. A. Sloane, Illustration for a(3) = 3

Cf. A100664, A108066, A348456.

It appears that a(8) = 8522. - Ian Henderson, Feb 07 2023, corrected Mar 07 2023

a(7) corrected by N. J. A. Sloane, Mar 07 2023, based on the second John Baez blog entry.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A100664 Number of inequivalent ways to dissect a square into n rectangles of equal perimeter.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A219861 Number of ways to dissect a nonsquare rectangle into n rectangles of equal area up to symmetry.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A359146 Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A257555 Number of distinct ways to dissect a cube into n rectangular boxes of equal volume, up to reflection and rotation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs