%I #33 Jan 21 2023 09:32:46

%S 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 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,

%U 51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68

%N 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 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 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 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 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 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 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 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 Stuart Anderson, <a href="http://www.squaring.net/sq/ss/spss/o21/spsso21.html">SPSS, Order 21</a>.

%H Stuart Anderson, <a href="http://www.squaring.net/sq/sc/spsc.html">Simple Perfect Square-Cylinders (SPSCs); Order 20</a>.

%H Stuart Anderson, <a href="http://www.squaring.net/sq/ss/quilts/mrs-perkins-quilts.html">Mrs Perkins's Quilt</a>.

%H 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 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 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 Hugo Pfoertner, <a href="/A342558/a342558_2.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, January 2023

%H Hugo Pfoertner, <a href="/A342558/a342558.pdf">Visualization of the resistor networks with n >= 13 using the Mathematica graph function</a>, January 2023

%H Rainer Rosenthal, <a href="/A342558/a342558_5.txt">Cascade graphs for examples n <= 21</a>, January 2023

%H Rainer Rosenthal, <a href="/A342558/a342558.png">Network and semi-quilt visualizing a(18)</a>, January 2023

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MrsPerkinssQuilt.html">Mrs. Perkins's Quilt</a>.

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

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

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

%K nonn

%O 1,6

%A _Hugo Pfoertner_ and _Rainer Rosenthal_, May 26 2021