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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.

%I A342558 #37 Jul 29 2025 17:25:55
%S A342558 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,
%T A342558 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,
%U A342558 51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68
%N 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.
%C A342558 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.
%C A342558 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".
%C A342558 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")
%C A342558 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).
%C A342558 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.
%C A342558 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.
%H A342558 Stuart Anderson, <a href="http://www.squaring.net/sq/ss/spss/spss.html">Simple Perfect Squared Squares (SPSSs)</a>, Order 21 to 37 and higher orders.
%H A342558 Stuart Anderson, <a href="http://www.squaring.net/sq/ss/spss/o21/spsso21.html">SPSS, Order 21</a>.
%H A342558 Stuart Anderson, <a href="http://www.squaring.net/sq/sc/spsc.html">Simple Perfect Square-Cylinders (SPSCs); Order 20</a>.
%H A342558 Stuart Anderson, <a href="http://www.squaring.net/sq/ss/quilts/mrs-perkins-quilts.html">Mrs Perkins's Quilt</a>.
%H A342558 A. Augusteijn and A. J. W. Duijvestijn, <a href="https://doi.org/10.1016/0095-8956(83)90061-8">Simple perfect square-cylinders of low order</a>, Journal of Combinatorial Theory, Series B, Volume 35, Issue 3, December 1983, Pages 333-337
%H A342558 A. J. W. Duijvestijn, <a href="https://doi.org/10.1016/0095-8956(78)90041-2">Simple perfect squared square of lowest order</a>, Journal of Combinatorial Theory, Series B, Volume 25, Issue 2, October 1978, Pages 240-243
%H A342558 Ed Pegg Jr., <a href="http://www.mathpuzzle.com/perkinsbestquilts.txt">List of solutions for the Mrs. Perkins's Quilt Square packing problem</a>.
%H A342558 Hugo Pfoertner, <a href="/A342558/a342558_6.txt">Examples of networks of n one-ohm-resistors with total resistance of 1 ohm, maximizing the number of distinct currents through the single resistors</a>, May 2021, Jan-Apr 2023
%H A342558 Hugo Pfoertner, <a href="/A342558/a342558_1.pdf">Visualization of the resistor networks with n >= 13 using the Mathematica graph function</a>, Jan-Apr 2023
%H A342558 Rainer Rosenthal, <a href="/A342558/a342558_5.txt">Cascade graphs for examples n <= 21</a>, January 2023
%H A342558 Rainer Rosenthal, <a href="/A342558/a342558.png">Network and semi-quilt visualizing a(18)</a>, January 2023
%H A342558 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MrsPerkinssQuilt.html">Mrs. Perkins's Quilt</a>.
%F A342558 a(n) = n for n >= 18.
%e A342558 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.
%Y A342558 Cf. A002962, A005670, A006983, A014530, A160911, A180414, A181340, A217156, A337517, A338593, A342556, A360030.
%K A342558 nonn
%O A342558 1,6
%A A342558 _Hugo Pfoertner_ and _Rainer Rosenthal_, May 26 2021