A014530 -id:A014530 - cp's OEIS frontend

A006983 Number of simple perfect squared squares of order n up to symmetry.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 12, 26, 160, 441, 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, 17086918

Offset: 1

N. J. A. Sloane

A squared rectangle (which may be a square) is a rectangle dissected into a finite number of two or more squares. If no two 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. - Geoffrey H. Morley, Oct 17 2012

J.-P. Delahaye, Les inattendus mathématiques, Belin-Pour la Science, Paris, 2004, pp. 95-96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stuart E. Anderson, Perfect Squared Rectangles and Squared Squares
Stuart E. Anderson, Simple perfect squared squares in orders 27 to 37 - methods used and people involved.
C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992) 67-75.
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992.
C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994.
C. J. Bouwkamp, A. J. W. Duijvestijn, & N. J. A. Sloane, Correspondence, 1971.
G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007), 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
A. J. W. Duijvestijn, Illustration for a(21)=1 (The unique simple squared square of order 21. Reproduced with permission of the discoverer.)
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of orders 21 to 24, J. Combin. Theory Ser. B 59 (1993), 26-34.
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332. doi:10.1090/S0025-5718-1994-1208220-9
A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. doi:10.1090/S0025-5718-96-00705-3 [TableI List of Simple Perfect Squared Squares of order 26 and TableII List of Simple Perfect Squared 2x1 Rectangles of order 26 are now on squaring.net and no longer located as described in the paper.]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 25.
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Index entries for squared squares

Cf. A002962, A002881, A002839, A014530.
Cf. A181340, A181735, A217155, A217156.
Cf. A129947, A217149, A228953 (related to sizes of the squares).
Cf. A349205, A349206, A349207, A349208, A349209, A349210 (related to ratios of element and square sizes).

Leading term changed from 0 to 1, Apr 15 1996
More terms from Stuart E Anderson, May 08 2003, Nov 2010
Leading term changed back to 0, Dec 25 2010 (cf. A178688)
a(29) added by Stuart E Anderson, Aug 22 2010; contributors to a(29) include Ed Pegg Jr and Stephen Johnson
a(29) changed to 7901, identified a duplicate tiling in order 29. - Stuart E Anderson, Jan 07 2012
a(28) changed to 3000, identified a duplicate tiling in order 28. - Stuart E Anderson, Jan 14 2012
a(28) changed back to 3001 after a complete recount of order 28 SPSS recalculated from c-nets with cleansed data, established the correct total of 3001. - Stuart E Anderson, Jan 24 2012
Definition clarified by Geoffrey H. Morley, Oct 17 2012
a(30) added by Stuart E Anderson, Apr 10 2013
a(31), a(32) added by Stuart E Anderson, Sep 29 2013
a(33), a(34) and a(35) added by Stuart E Anderson, May 02 2016
Moved comments on orders 27 to 35 to a linked file.  Stuart E Anderson, May 02 2016
a(36) and a(37) enumerated by Jim Williams, added by Stuart E Anderson, Jul 26 2020.

A217156 Number of perfect squared squares of order n up to symmetries of the square.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 12, 30, 172, 541, 1372, 3949, 10209, 26234, 71892, 196357, 528866, 1420439, 3784262, 10012056, 26048712

Offset: 1

Geoffrey H. Morley, Sep 27 2012

a(n) is the number of solutions to the classic problem of 'squaring the square' by n unequal squares. 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, and compound if it does.

			a(21) = 1 because there is a unique perfect squared square of order 21. A014530 gives the sizes of its constituent squares.

H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991, section C2, pp. 81-83.
A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
P. J. Federico, Squaring rectangles and squares: A historical review with annotated bibliography, in Graph Theory and Related Topics, J. A. Bondy and U. S. R. Murty, eds., Academic Press, 1979, 173-196.
J. H. van Lint and R. M. Wilson, A course in combinatorics, Chapter 34 "Electrical networks and squared squares", pp. 449-460, Cambridge Univ. Press, 1992.
J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.
I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.
W. T. Tutte, Squaring the Square, in M. Gardner's 'Mathematical Games' column in Scientific American 199, Nov. 1958, pp. 136-142, 166. Reprinted with addendum and bibliography in the US in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, Simon and Schuster, New York (1961), pp. 186-209, 250, and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-7.
W. T. Tutte, Graph theory as I have known it, Chapter 1 "Squaring the square", pp. 1-11, Clarendon Press, Oxford, 1998.

S. E. Anderson, Simple Perfect Squared Squares (complete to order 29).
S. E. Anderson, Compound Perfect Squared Squares (complete to order 29).
J. A. Bondy and U. S. R. Murty, Chapter 12: The Cycle Space and Bond Space, pp. 212-226 in: Graph theory with applications, Elsevier Science Ltd/North-Holland, 1976.
C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992), 67-75.
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992.
C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994.
G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007), 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340. Reprinted in I. Gessel and G.-C. Rota (editors), Classic papers in combinatorics, Birkhäuser Boston, 1987, pp. 88-116.
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep. 17 (1962), 523-612.
A. J. W. Duijvestijn, P. J. Federico, and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 15-32. [The lowest order of a compound perfect square is 24.]
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of orders 21 to 24, J. Combin. Theory Ser. B 59 (1993), 26-34.
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332.
A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. [TableI List of Simple Perfect Squared Squares of order 26 and TableII List of Simple Perfect Squared 2x1 Rectangles of order 26 are now on squaring.net and no longer located as described in the paper.]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 25.
C. A. B. Smith and W. T. Tutte, A class of self-dual maps, Can. J. Math., 2 (1950), 179-196.
W. T. Tutte, Squaring the square, Can. J. Math., 2 (1950), 197-209.
W. T. Tutte, The quest of the perfect square, Amer. Math. Monthly 72 (1965), 29-35.
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Wikipedia, Squaring the square
Index entries for squared squares

Cf. A181735 (counts symmetries of any squared subrectangles as equivalent).

Cf. A110148, A217154.

a(n) = A006983(n) + A217155(n).

Added a(29) = 10209, Stuart E Anderson, Nov 30 2012
Added a(30) = 26234, Stuart E Anderson, May 26 2013
Added a(31) = 71892, a(32) = 196357, Stuart E Anderson, Sep 30 2013
Added a(33) = 528866, a(34) = 1420439, a(35) = 3784262, due to enumeration completed by Jim Williams in 2014 and 2016. Stuart E Anderson, May 02 2016
a(36) and a(37) completed by Jim Williams in 2016 to 2018, added by Stuart E Anderson, Oct 28 2020

A181735 Number of perfect squared squares of order n up to symmetries of the square and of its squared subrectangles, if any.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 12, 27, 162, 457, 1198, 3144, 8313, 21507, 57329, 152102, 400610, 1053254, 2750411, 7140575, 18326660

Offset: 1

Stuart E Anderson

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, and compound if it does. - Geoffrey H. Morley, Oct 17 2012

			From _Geoffrey H. Morley_, Oct 17 2012 (Start):
a(21) = 1 because there is a unique perfect squared square of order 21. A014530 gives the sizes of its constituent squares.
a(24) = 27 because there are A217156(24) = 30 perfect squared squares of order 24 but four of them differ only in the symmetries of a squared subrectangle. (End)

See A217156 for further references and links.
J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.

S. E. Anderson, Simple Perfect Squared Squares (complete to order 29).
S. E. Anderson, Compound Perfect Squared Squares (complete to order 28).
C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992), 67-75. doi:10.1016/0012-365X(92)90531-J
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992.
C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994.
G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007), 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
A. J. W. Duijvestijn, P. J. Federico and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 15-32. [The lowest order of a compound perfect square is 24.]
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of orders 21 to 24, J. Combin. Theory Ser. B 59 (1993), 26-34.
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332. doi:10.1090/S0025-5718-1994-1208220-9
A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. doi:10.1090/S0025-5718-96-00705-3 [TableI List of Simple Perfect Squared Squares of order 26 and TableII List of Simple Perfect Squared 2x1 Rectangles of order 26 are now on squaring.net and no longer located as described in the paper.]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 25. [But symmetries of any subrectangles counted as distinct.]
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Wikipedia, Squaring the square
Index entries for squared squares

Cf. A217156 (counts symmetries of any subrectangles as distinct).

Cf. A110148, A217154.

a(n) = A006983(n) + A181340(n). - Geoffrey H. Morley, Oct 17 2012

Corrected last term to 3144 to reflect correction to 143 of last order 28 compound squares term in A181340.
Added more clarification in comments on definition of a perfect squared square. - Stuart E Anderson, May 23 2012
Definition corrected and offset changed to 1 by Geoffrey H. Morley, Oct 17 2012
a(29) added by Stuart E Anderson, Dec 01 2012
a(30) added by Stuart E Anderson, May 26 2013
a(31) and a(32) added by Stuart E Anderson, Sep 30 2013
a(33), a(34) and a(35) added after enumeration by Jim Williams, Stuart E Anderson, May 02 2016
a(36) and a(37) from Jim Williams, completed in 2018 to 2020, added by Stuart E Anderson, Oct 28 2020

A002962 Number of simple imperfect squared squares of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 5, 15, 19, 57, 72, 274, 491, 1766, 3679, 11158, 24086, 64754, 132598, 326042, 667403, 1627218, 3508516

Offset: 1

N. J. A. Sloane

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. 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. - Geoffrey H. Morley, Oct 17 2012

Orders 15 to 19 were enumerated by C. J. Bowkamp and A. J. W. Duijvestijn (1962). Orders 20 to 29 were enumerated by Stuart Anderson (2010-2012). Orders 30 to 32 were enumerated by Lorenz Milla and Stuart Anderson (2013). - Stuart E Anderson, Sep 30 2013

C. J. Bouwkamp, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stuart Anderson, Simple Imperfect Squared Squares.
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep., 17 (1962), 523-612. [Pp. 573-4 have simple imperfect squares up to order 19.]
Index entries for squared squares

Cf. A002839, A002881, A006983, A014530.

Cf. A181735, A217155, A217156.

a(19) corrected and terms extended up to a(22) by Stuart E Anderson, Mar 08 2011
a(21) and a(22) corrected and terms extended to a(25) by Stuart E Anderson, Apr 24 2011
a(21), a(22), a(25) corrected and a(26)-a(28) added by Stuart E Anderson, Jul 11 2011
a(29) from Stuart E Anderson, Ed Pegg Jr, Stephen Johnson, Aug 22 2011
a(29) corrected by Stuart E Anderson, Aug 24 2011
Definition clarified and offset changed to 1 by Geoffrey H. Morley, Oct 17 2012
a(28) corrected by Stuart E Anderson, Dec 01 2012
a(30) from Lorenz Milla and Stuart E Anderson, Apr 10 2013
a(26) and a(29) corrected by Stuart E Anderson, Aug 20 2013
a(31), a(32) from Lorenz Milla and Stuart E Anderson, Sep 30 2013

A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, May 26 2021

nonn

The resistor networks considered here correspond to multigraphs in which each edge is replaced by one or more one-ohm resistors, and in which there are two distinguished nodes, called poles, between which there is a total resistance of 1 ohm.
It was known that the smallest resistor network with all currents being distinct consists of 21 resistors, found by Duijvestin in 1978. This assumes that the network is planar and thus the analogy to the perfectly tiled squares exists, see A014530. For history and references see link to Stuart Anderson's website "SPSS, Order 21".
In 1983, A. Augusteijn and A. J. W. Duijvestijn  described networks in which the number of resistors in a network with distinct resistances was reduced to 20 by allowing the tiled square to be wrapped onto a cylinder. (see links to their publication and to Stuart Anderson's website "Simple Perfect Square-Cylinders")
For values of n greater than 21 increasingly numerous square divisions with a(n) = n exist so that a(n) = n holds for all n > 21 (see A006983).
In the present sequence, networks based on non-planar graphs are allowed, which makes it possible to find networks with a(n) = n also for n = 18 and n = 19.
In the range from n = 13 to n = 17, larger numbers of distinct currents are found than are possible with the methods for generating Mrs. Perkins's quilts, which naturally correspond to planar graphs.

			Examples for n <= 21 are given in the Pfoertner links. Visualizations of tilings corresponding to optimal networks for n <= 12 are given in the Mathworld "Mrs. Perkins's Quilt" link.

Stuart Anderson, Simple Perfect Squared Squares (SPSSs), Order 21 to 37 and higher orders.
Stuart Anderson, SPSS, Order 21.
Stuart Anderson, Simple Perfect Square-Cylinders (SPSCs); Order 20.
Stuart Anderson, Mrs Perkins's Quilt.
A. Augusteijn and A. J. W. Duijvestijn, Simple perfect square-cylinders of low order, Journal of Combinatorial Theory, Series B, Volume 35, Issue 3, December 1983, Pages 333-337
A. J. W. Duijvestijn, Simple perfect squared square of lowest order, Journal of Combinatorial Theory, Series B, Volume 25, Issue 2, October 1978, Pages 240-243
Ed Pegg Jr., List of solutions for the Mrs. Perkins's Quilt Square packing problem.
Hugo Pfoertner, Examples of networks of n one-ohm-resistors with total resistance of 1 ohm, maximizing the number of distinct currents through the single resistors, May 2021, Jan-Apr 2023
Hugo Pfoertner, Visualization of the resistor networks with n >= 13 using the Mathematica graph function, Jan-Apr 2023
Rainer Rosenthal, Cascade graphs for examples n <= 21, January 2023
Rainer Rosenthal, Network and semi-quilt visualizing a(18), January 2023
Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt.

Cf. A002962, A005670, A006983, A014530, A160911, A180414, A181340, A217156, A337517, A338593, A342556, A360030.

a(n) = n for n >= 18.

A340726 Maximum power V_s*A_s consumed by an electrical network with n unit resistors and input voltage V_s and current A_s constrained to be exact integers which are coprime, and such that all currents between nodes are integers.

Original entry on oeis.org

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 13130, 41591, 118590, 404471, 1158696, 3893831, 12222320, 39428991, 123471920, 397952081, 1297210320

Offset: 1

Rainer Rosenthal, Jan 17 2021

This sequence is an analog of A338861. Equality a(n) = A338861(n) holds for small n only, see example.
Let V_s denote the specific voltage, i.e., the lowest integer voltage, which induces integer currents everywhere in the network. Denote by A_s the specific current, i.e., the corresponding total current.
A planar network with n unit resistors corresponds to a squared rectangle with height V_s and width A_s. The electrical power V_s*A_s therefore equals the area of that rectangle. In the historical overview (Stuart Anderson link) A_s is called complexity.
The corresponding rectangle tiling provides the optimal power rating of the 1 ohm resistors with respect to the specific voltage V_s and current A_s. See the picture From_Quilt_to_Net in the link section, which also provides insight in the "mysterious" correspondence between rectangle tilings and electric networks. For non-planar nets the idea of rectangle tilings can be widened to 'Cartesian squarings'. A Cartesian squaring is the dissection of the product P X Q of two finite sets into 'squaresets', i.e., sets A X B with A subset of P and B subset of Q, and card(A) = card(B). - Rainer Rosenthal, Dec 14 2022
Take the set SetA337517(n) of resistances, counted by A337517. For each resistance R multiply numerator and denominator. Conjecture: a(n) is the maximum of all these products. The reason is that common factors of V_s and A_s are quite rare (see the beautiful exceptional example with 21 resistors).

			n = 3:
Networks with 3 unit resistors have A337517(3) = 4 resistance values: {1/3, 3, 3/2, 2/3}. The maximum product numerator X denominator is 6.
n = 6:
Networks with 6 unit resistors have A337517(6) = 57 resistance values, where 11/13 and 13/11 are the resistances with maximum product numerator X denominator.
                                             +-----------+-------------+
                     A                       |           |             |
                    / \                      |           |             |
               (1) /   \ (2)                 |   6 X 6   |    7 X 7    |
                  /     \                    |           |             |
                 /  (3)  \                   |           |             |
                o---------o                  +---------+-+             |
                 \       //                  |         +-+-----+-------+
                  \  (5)//                   |  5 X 5  |       |       |
               (4) \   //(6)                 |         | 4 X 4 | 4 X 4 |
                    \ //                     |         |       |       |
                     Z                       +---------+-------+-------+
       ___________________________________________________________________
        Network with 6 unit resistors       Corresponding rectangle tiling
        total resistance 11/13 giving          with 6 squares giving
            a(6) = 11 X 13 = 143                 A338861(6) = 143
n = 10:
With n = 10, non-planarity comes in, yielding a(10) > A338861(10).
The "culprit" here is the network with resistance A338601(9)/A338602(9) = 130/101, giving a(10) = 13130 > A338861(10) = 10920.
n = 21:
The electrical network corresponding to the perfect squared square A014530 has specific voltage V_s equal to specific current A_s, namely V_s = A_s = 112. Its power V_s*A_s = 12544 is far below the maximum a(20) > a(10) > 13000, and a(n) is certainly monotonically increasing. - _Rainer Rosenthal_, Mar 28 2021

Rainer Rosenthal, From_Quilt_to_Net
Squaring.Net 2020, Stuart Anderson, Squared Rectangle and Smith Diagram
Index to sequences related to resistances.

Cf. A180414, A337517, A338601, A338602, A338861.

a(13)-a(17) from Hugo Pfoertner, Feb 08 2021
Definition corrected by Rainer Rosenthal, Mar 28 2021
a(18) from Hugo Pfoertner, Apr 09 2021
a(19)-a(20) from Hugo Pfoertner, Apr 16 2021

A217150 Smallest number, two or more, of unequal squares that tile a square in exactly n ways; or 0 if there is no such set of tiles.

Original entry on oeis.org

21, 25, 28, 24

Offset: 1

Geoffrey H. Morley, Sep 27 2012

The definition precludes the trivial tiling by only one square.

a(8) = 25, a(16) = 26, a(32) = 28, and a(48) = 28. For all other n > 4, a(n) > 29 or a(n) = 0. a(n) > 29 (with a known upper bound) for n = 7, 24, 56, 64, 72, 96, 112, 128 ...

			See MathWorld link for an explanation of Bouwkamp code used in these examples.
a(2) = 25. The 25 squares of the following perfect square with side 540 can be arranged in one other way by rearranging polygons a-c: (279,261)(98,68,95)(135a,144a)(30,38)(11,84)(55,65,8)(57)(126b,9a)(45c,10)(117c,36c)(116,16)(100)(81c).
a(3) = 28. The 28 squares of the following perfect square with side 408 can be arranged in two other ways by rearranging polygons a-e: (165,102a,141a)(63a,39a)(24a,156b)(99c,61c,92c)(38c,23c)(15c,8c)(7c)(9c,20c,64c)(144c,13c,2c)(11c)(44c)(45d,111d)(87e,21d)(66d). No other set of fewer than 30 unequal squares tiles a square in exactly three ways.

C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992), 67-75.
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332.
Eric Weisstein's World of Mathematics, Perfect Square Dissection

Cf. A014530, A217151.

cp's OEIS Frontend

A006983 Number of simple perfect squared squares of order n up to symmetry.

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A217156 Number of perfect squared squares of order n up to symmetries of the square.

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A181735 Number of perfect squared squares of order n up to symmetries of the square and of its squared subrectangles, if any.

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A002962 Number of simple imperfect squared squares of order n up to symmetry.

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A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

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A340726 Maximum power V_s*A_s consumed by an electrical network with n unit resistors and input voltage V_s and current A_s constrained to be exact integers which are coprime, and such that all currents between nodes are integers.

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A217150 Smallest number, two or more, of unequal squares that tile a square in exactly n ways; or 0 if there is no such set of tiles.

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A340919 Sorted sizes of the 55 squares used in the first known squared square of dimension 4205 X 4205 found by Roland Sprague in 1938.

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