A181735 -id:A181735 - cp's OEIS frontend

A217148 Smallest possible side length for a perfect squared square of order n; or 0 if no such square exists.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 112, 110, 110, 120, 147, 212, 180, 201, 221, 201, 215, 185, 233, 218, 225, 253, 237

Offset: 1

Geoffrey H. Morley, Sep 27 2012

It is not known whether this sequence is the same as sequence A129947. It may be that A129947(33) = 246 and A217148(33) = 234. - Geoffrey H. Morley, Jan 10 2013
A squared rectangle (which may be a square) 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.
The upper bounds shown below for 38 and 40-44 are from J. B. Williams. Those for n = 39 and 45-47 are from Gambini's thesis. - Geoffrey H. Morley, Mar 08 2013
======================================
Upper bounds for a(n) for n = 31 to 59
======================================
   |  +0   +1   +2   +3   +4   +5   +6   +7   +8   +9
======================================================
30 |   -    -    -    -    -    -    -    -   352  360
40 |  328  336  360  413  425  543  601  691  550  583
50 |  644  636  584  685  657  631  751  742  780  958
======================================================
The sequence A129947 has identical terms to A217148 (so far), however they are different as A129947 refers to simple perfect squared squares (SPSSs), while A217148 refers to SPSSs and compound perfect squared squares (CPSSs).  The simples and compounds together are referred to as perfect squared squares (PSSs).  So far it has been observed that all the smallest side lengths belong to SPSSs only. - Stuart E Anderson, Oct 27 2020

S. E. Anderson, Perfect Squared Rectangles and Squared Squares.
Stuart E. Anderson, 'Special' Perfect Squared Squares", accessed 2014. - _N. J. A. Sloane_, Mar 30 2014
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, pp. 73-78.
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection

Cf. A129947, A174386, A181735, A217149, A217156.

a(29) from Stuart E Anderson added by Geoffrey H. Morley, Nov 23 2012
a(30) from Stuart E Anderson and Lorenz Milla added by Geoffrey H. Morley, Jun 15 2013
a(31) and a(32) from Lorenz Milla and Stuart E Anderson, Oct 05 2013
For additional terms see the Ed Pegg link, also A006983. - N. J. A. Sloane, Jul 29 2020
a(33) to a(37) from J. B. Williams added by Stuart E Anderson, Oct 27 2020

A219766 Number of nonsquare simple perfect squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931307, 17633765, 63301415, 228130900, 825228950, 2994833413

Offset: 1

Stuart E Anderson, Nov 27 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 02 2024]

See A006983 and A217156 for references.

Stuart E Anderson, Simple Perfect Squared Rectangles. [Nonsquare rectangles only]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
See A006983 and A217156 for further links.

Cf. A002839, A006983, A002962, A002881, A181735.

Cf. A217153, A217154, A217156.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A002839 = A@002839;
A006983 = A@006983;
a[n_] := A002839[[n]] - A006983[[n]];
a /@ Range[24] (* Jean-François Alcover, Jan 13 2020 *)

a(n) = A002839(n) - A006983(n).
In "A Census of Planar Maps", p. 267, William Tutte gave a conjectured asymptotic formula for the number, a(n) of perfect squared rectangles of order n:
Conjectured: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). [Corrected by Stuart E Anderson, Feb 02 2024]

a(9)-a(24) enumerated by Gambini 1999, confirmed by Stuart E Anderson, Dec 07 2012
a(25) from Stuart E Anderson, May 07 2024
a(26) from Stuart E Anderson, Jul 28 2024

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

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A217148 Smallest possible side length for a perfect squared square of order n; or 0 if no such square exists.

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A219766 Number of nonsquare simple perfect squared rectangles of order n up to symmetry.

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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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