A002839 -id:A002839 - cp's OEIS frontend

A220165 Number of nonsquare simple imperfect squared rectangles of order n up to symmetry.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 9, 33, 104, 280, 948, 3014, 9494, 30302, 98897, 323372, 1080168, 3666666, 12604812, 43734613, 153788715

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares.

See A002881 and A006983.

Stuart E. Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24

Cf. A002881, A006983, A002962, A002839.

a(25) from Stuart E Anderson, May 07 2024

a(26) from Stuart E Anderson, Jul 28 2024

A217374 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle and its subrectangles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 16, 60, 194, 622, 2128, 7438, 25852, 90266, 317350, 1127800

Offset: 1

Geoffrey H. Morley, Oct 02 2012

A squared rectangle is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.

A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, Paper I, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); Paper II, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); Paper III, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63). [Paper I has terms up to a(13) on p. 1178.]
C. J. Bouwkamp, On the construction of simple perfect squared squares, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625).
Index entries for squared rectangles

Cf. A217375 (counts symmetries of squared subrectangles as distinct).

Cf. A110148.

a(n) = a(n-1) + 2*A002839(n-1) + 2*A217152(n-1).

a(20) corrected by Geoffrey H. Morley, Oct 12 2012

A217375 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 40, 168, 604, 2076, 7320, 26132, 93352, 333992, 1199716, 4329180

Offset: 1

Geoffrey H. Morley, Oct 02 2012

A squared rectangle is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.

A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24. [A217153 up to a(18).]
Index entries for squared rectangles

Cf. A217374 (counts symmetries of squared subrectangles as equivalent).

Cf. A217154.

a(n) >= 2*a(n-1) + 4*A002839(n-1) + 4*A217153(n-1), with equality for n<19.

a(20) corrected by Geoffrey H. Morley, Oct 12 2012

A334905 a(n) is the minimum remaining space when a square n X n is tiled with smaller squares with distinct integer sides parallel to the n X n square.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 21, 30, 29, 20, 25, 30, 12, 19, 24, 17, 13, 13, 18, 14, 19, 14, 15, 15, 15, 20, 15, 20, 16, 22, 16, 16, 17, 21, 22, 15, 13, 16, 18, 14, 14, 14, 17, 15, 11, 10, 12, 13, 4, 11, 8, 9, 7, 11, 4, 9, 8, 8, 8, 6, 8

Offset: 1

Vitor Pimenta dos Reis Arruda, May 15 2020

nonn

See (Gambini, 1999) for a way to construct the sequence. Actually, one would have to extend Gambini's idea by putting extra 1-sided squares in the list of "usable squares" to allow finding nonzero-waste packings.

			For n=5, squares of sides {1, 4} can be packed inside the container, leading to uncovered area a(5) = 5*5 - (4*4 + 1*1) = 8. The other maximal packable set is composed of the squares sided {1,2,3}, which would lead to uncovered area greater than 8.

Vitor Pimenta dos Reis Arruda, Table of n, a(n) for n = 1..101
I. Gambini, A method for cutting squares into distinct squares, Discrete Applied Mathematics, 98 (1999), 65-80.
Vitor Pimenta dos Reis Arruda, Non trivial decompositions until a(101)
Vitor Pimenta dos Reis Arruda, Luiz Gustavo Bizarro Mirisola, and Nei Yoshihiro Soma, Almost squaring the square: optimal packings for non-decomposable squares, Pesqui. Oper. (2022) Vol. 42.
Giovanni Resta, Illustration of terms a(15)-a(31)
Wikipedia, Squaring the square

Cf. A006983, A002962, A181735, A002839, A228953.

Terms a(17)-a(31) from Giovanni Resta, May 15 2020

A195984 The size of the smallest boundary square in simple perfect squared rectangles of order n.

Original entry on oeis.org

8, 13, 22, 18, 14, 13, 11, 9, 6, 9, 7, 7, 8, 6, 8, 7

Offset: 9

Stuart E Anderson, Sep 26 2011

nonn

Ian Gambini showed in his thesis that the minimum value for a(n) is 5. Brian Trial found 3 simple perfect squared rectangles (SPSRs) of order 28 with boundary squares of size 5 in September 2011. An unsolved problem is to find the lowest order SPSR with a '5 on the side'.
Added a(22) = 6 (Stuart Anderson), Brian Trial has found a(28) = 5. This gives an upper bound of 28, in addition to the lower bound of 23, to the problem of finding the lowest order SPSR with a square of size 5 on the boundary. - Stuart E Anderson, Sep 29 2011
Found a(23) = 8, the lower bound is now order 24. - Stuart E Anderson, Nov 30 2012
Found a(24) = 7, the lower bound is now order 25. - Stuart E Anderson, Dec 07 2012

Gambini, Ian. Thesis; 'Quant aux carrés carrelés' L’Universite de la Mediterranee Aix-Marseille II 1999

Stuart E. Anderson, Simple Perfects by Boundary Rules and Conditions
Stuart Anderson, 'Special' Perfect Squared Squares", accessed 2014. - _N. J. A. Sloane_, Mar 30 2014

Cf. A002839.

Added a(23) = 8, Stuart E Anderson, Nov 30 2012

Added a(24) = 7, Stuart E Anderson, Dec 07 2012

A220166 Number of nonsquare simple squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 6, 22, 76, 246, 848, 2889, 9964, 34440, 119875, 420525, 1482802, 5254679, 18713933, 66968081, 240735712

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of integer sized squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. This sequence counts nonsquare simple perfect squared rectangles and nonsquare simple imperfect squared rectangles.

See A006983 and A217156 for further links.

S. E. Anderson, Simple Perfect Squared Rectangles [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Rectangles [Nonsquare rectangles only]

Cf. A219766, A220165, A002839, A002881.

a(9)-a(24) from Stuart E Anderson Dec 07 2012

A220167 Number of simple squared rectangles of order n up to symmetry.

Original entry on oeis.org

3, 6, 22, 76, 247, 848, 2892, 9969, 34455, 119894, 420582, 1482874, 5254954, 18714432, 66969859, 240739417

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle is a rectangle dissected into a finite number of integer-sized squares. If no two of these squares are the same size then the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle or squared square. The order of a squared rectangle is the number of squares into which it is dissected. [Edited by Stuart E Anderson, Feb 03 2024]

See A006983 and A217156 for references and links.

S. E. Anderson, Simple Perfect Squared Rectangles. [Nonsquare rectangles only]
S. E. Anderson, Simple Perfect Squared Squares.
S. E. Anderson, Simple Imperfect Squared Rectangles. [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Squares.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Cf. A002839, A002881, A006983, A002962, A220165, A219766.

a(n) = A002839(n) + A002881(n).
a(n) = A006983(n) + A002962(n) + A220165(n) + A219766(n).
Conjecture: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)), from "A Census of Planar Maps", p. 267, where William Tutte gave a conjectured asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order). [Corrected by Stuart E Anderson, Feb 03 2024]

a(9)-a(24) from Stuart E Anderson, Dec 07 2012

cp's OEIS Frontend

A220165 Number of nonsquare simple imperfect squared rectangles of order n up to symmetry.

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A217374 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle and its subrectangles.

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A217375 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle.

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A334905 a(n) is the minimum remaining space when a square n X n is tiled with smaller squares with distinct integer sides parallel to the n X n square.

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A195984 The size of the smallest boundary square in simple perfect squared rectangles of order n.

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A220166 Number of nonsquare simple squared rectangles of order n up to symmetry.

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A220167 Number of simple squared rectangles of order n up to symmetry.

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