A160911 -id:A160911 - cp's OEIS frontend

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.

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.

A210517 Number of rectangles dissectible into n squares, unique up to aspect ratio.

Original entry on oeis.org

1, 1, 2, 5, 11, 28, 74, 211

Offset: 1

Geoffrey H. Morley, Jan 26 2013

The rectangles are distinguishable by aspect ratio, not size.

A rectangle is dissectible into squares if and only if its sides are commensurable. A rectangle with commensurable sides is dissectible into n squares for all but a finite number of positive integers n. For example, a square is dissectible into any number of squares other than 2, 3, or 5.

			For n = 3 the a(3) = 2 rectangles are 3 X 1 and 3 X 2 with aspect ratio 3/1 and 3/2. For example, a 3 X 2 rectangle can be tiled by a 2 X 2 square and two 1 X 1 squares.
For n = 4 the a(4) = 5 aspect ratios are 1/1, 4/1, 4/3, 5/2 and 5/3. Ratio 1/1 stems from the square 2 X 2, tiled by four 1 X 1 squares.
For n = 5 the a(5) = 11 aspect ratios are 2/1, 5/1, 5/4, 6/5, 7/2, 7/3, 7/4, 7/5, 7/6, 8/3 and 8/5.
For n = 6 the a(6) = 28 aspect ratios are 1/1, 3/1, 3/2, 4/3, 5/4, 6/1, 6/5, 9/2, 9/4, 9/5, 9/7, 10/3, 10/7, 10/9, 11/3, 11/4, 11/5, 11/6, 11/7, 11/8, 11/10, 12/5, 12/7, 13/5, 13/6, 13/7, 13/8 and 13/11.

Rainer Rosenthal, SetA210517(n) = set of aspect ratios for n squares (for n < 9).

Cf. A160911 (tilings with same aspect ratio allowed), A221839.

Title changed by Rainer Rosenthal, Dec 30 2022

a(7) corrected, a(8) new. - Marx Stampfli and Rainer Rosenthal, Jan 10 2023

A373692 Table of the number of ways T(m,n) to partition a 2m X 2n grid into Cartesian products of size 2 X 2, read by ascending antidiagonals.

Original entry on oeis.org

1, 3, 3, 15, 45, 15, 105, 1575, 1575, 105, 945, 99225, 510525, 99225, 945, 10395, 9823275, 376473825, 376473825, 9823275, 10395, 135135, 1404728325, 533407191975, 4202869719825, 533407191975, 1404728325, 135135, 2027025, 273922023375, 1302400497234375, 115509334438258425, 115509334438258425, 1302400497234375, 273922023375, 2027025

Offset: 1

Rainer Rosenthal and Markus Sigg, Jun 13 2024

This is a special case of the problem to partition a Cartesian product P X Q into squares P_i X Q_i, i.e. |P_i| = |Q_i|. In our case all subsets have size 2. Using the terminology of A160911 we deal with partitions of type (2m X 2n: 2,2,2,...).
From Markus Sigg, Jul 25-26 2024: (Start)
T(m,n) is a multiple of (2m-1)(2n-1) as there are that many ways to place a Cartesian product with one point in the top left of the grid, and the resulting configurations are equivalent.
For m,n > 1, starting with the Cartesian product {1,2m} X {1,2n} and evaluating the options for adding a Cartesian product with one point in (1,2) shows that T(m,n) is a multiple of (2m-1)(2n-1)*lcm(2m-3,2n-3). (End)

			Table T(m,n) begins:
.
     n       1             2                 3              4             5
  m \ ---------------------------------------------------------------------
  1 |        1             3                15            105           945
  2 |        3            45              1575          99225       9823275
  3 |       15          1575            510525      376473825  533407191975
  4 |      105         99225         376473825  4202869719825
  5 |      945       9823275      533407191975  115509334438258425
  6 |    10395    1404728325  1302400497234375  6907197292027901339625
  7 |   135135  273922023375
  8 |  2027025
.
These are the T(1,2) = 3 possible partitions:
.
    |A A B B|   |A B A B|   |A B B A|
    |A A B B|   |A B A B|   |A B B A|
    _________________________________
       #1          #2          #3
.
For T(2,2) = 45 consider these special partitions:
.
   |A A B B|   |A A B B|   |A A B B|
   |A A B B|   |A A B B|   |A A C C|
   |C C D D|   |C D C D|   |D D B B|
   |C C D D|   |C D C D|   |D D C C|
  ___________________________________
     Base1       Base2       Base3
.
Any partition is equivalent to exactly one of these partitions, i.e. it differs only by the order of the rows and columns. The number of equivalent partitions is respectively 9, 18, 18. Thus we have T(2,2) = 9 + 18 + 18 = 45.
See the picture and the expanded example in the link section.
.
Some other known terms: T(5,5) = 84250218148544569727025, T(6,4) = 6907197292027901339625, T(7,4) = 814287280679532017261528625, T(8,4) = 173936355367823940296258779550625, T(9,4) = 62626268302216078023651174787170095625, T(10,4) = 35784629301848063975515694953866493243805625.

Markus Sigg, Table of n, a(n) for n = 1..59
Rainer Rosenthal, Colored illustration showing that T(2,2) = 45.
Rainer Rosenthal, Expanded Example.
Markus Sigg, C program for computing the very first terms of the sequence.
Markus Sigg, C program for computing more terms of the sequence, namely T(ROWS/2,COLS/2) where ROWS and COLS are defined in the first program lines.

Cf. A001147 (column 1), A079484 (column 2 - conjectured), A160911.

C
```
// See Markus Sigg link.
```

a(24) and beyond from Markus Sigg, Jul 18 2024

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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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A210517 Number of rectangles dissectible into n squares, unique up to aspect ratio.

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A373692 Table of the number of ways T(m,n) to partition a 2m X 2n grid into Cartesian products of size 2 X 2, read by ascending antidiagonals.

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