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%I #54 Dec 17 2023 17:37:54
%S 95,101,98,97,103,97,110,103,130,103,115,106,109,98,101
%N Numerators x of resistance values R=x/y that can be obtained by a network of at most 10 one-ohm resistors such that a network of more than 10 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338602.
%C The terms are sorted by increasing value of the resistance R(n) = a(n)/A338602(n).
%C For more information, references, and links, see A180414 and A338573.
%C Each network for R = p/q is visualized (see link section) as a multigraph with the battery nodes on top and at the bottom, i.e., the battery edge does not have to cross any edges. Any planar network with 10 resistors presented in this way, has a corresponding tiling of a p X q rectangle by 10 squares, and the inverse resistance q/p can be obtained in the same way. According to the definition of a(n) this is not the case here, so there must be crossing edges in every drawing. It should be noticed though, that all the networks (without the battery edge) are planar. - _Rainer Rosenthal_, Jan 03 2021
%C Version 2 of the visualization (see link section) shows that all these exceptional networks are extensions of the same network with 8 resistors. It is the graph K_3_3 without the 'battery edge' A-Z and shall be named VG8. This network VG8 has no related squared rectangle, because it has no series-parallel subnets and has resistance 5/4, but there is no such network with resistance 4/5. So, this is the graph, which is mentioned by Karnofsky in his "Addendum": "The smallest non-planar graph has eight resistors.". - _Rainer Rosenthal_, Feb 13 2021
%C The reciprocal 101/130 of R(9) = 130/101 needs 12 resistors, while the other 14 reciprocal resistance values can be obtained by networks of 11 resistors. - _Rainer Rosenthal_, Jan 16 2021
%H Allan Gottlieb, <a href="https://cs.nyu.edu/~gottlieb/tr/overflow/2003-oct-3-more.html">Oct 3, 2003 addendum (Karnofsky)</a>.
%H Rainer Rosenthal, <a href="/A338601/a338601_2.png">Karnofsky's 15 exceptional networks with 10 resistors (Version 2)</a>.
%H <a href="/index/Res#resistances">Index to sequences related to resistances</a>.
%e All fractions for 10 resistors are: 95/106, 101/109, 98/103, 97/98, 103/101, 97/86, 110/91, 103/83, 130/101, 103/80, 115/89, 106/77, 109/77, 98/67, 101/67.
%e The corresponding networks are shown below, with -(always 1) and +(maximum node number) indicating the nodes where the voltage is applied. Edges marked ==, ||, //, or \\, have 2 resistors in parallel.
%e .
%e 95/106 101/109 98/103 97/98 103/101
%e -1=======2 -1-------2 -1-------2 -1-------2 -1-------2
%e |\ /| |\ /|| |\ /| |\ /| |\ /|
%e | \ / | | \ / || | \ / | | \ / | | \ / |
%e | \ / | | \ / || | \ / | | \ / | | \ / |
%e | 4 | | 4 || | 4 | | 6 4 4 6 |
%e | / \ | | / \ || | //\ | | / \ | | / \ |
%e | / +6 | | / +6 || | // +6 | | / +7 | | / +7 |
%e |/ \| |/ \|| |// \| |/ \| |/ \|
%e 3-------5 3-------5 3-------5 3-------5 3-------5
%e .
%e 97/86 110/91 103/83 130/101 103/80
%e -1=======2 -1 -1-------2 -1-----2 -1=======2
%e | /| / \ | /|| | /|\ | /|
%e | / | / \ | 4 || | | | | | 4 |
%e | / | 2-----3 | / || | | | | | /| |
%e | 6 | ||\ / \ | 6 || | 4-6 | | / 6 |
%e | / \ | || \4/ | | / \ || | / | | | / | |
%e | 4 +7 | || \ | | / +7 || | / +7 | | / +7 |
%e |/ \| \\ +6--5 |/ \|| |/ \| |/ \|
%e 3-------5 \\===// 3-------5 3-------5 3-------5
%e .
%e 115/89 106/77 109/77 98/67 101/67
%e -1 -1-------2 -1-------2 -1-------2 -1-------2
%e / \ | /|| | //| | /| | /|
%e / \ | 4 || | 4 | | 6 | | 6 |
%e 2-----3 | /| || | /| | | /| | | /| |
%e |\ / \ | / 6 || | / 6 | | / 7 | | / 7 4
%e | \6/ | | / | || | / | | | 4 | | | / | |
%e | \ | | / +7 || | / +7 | | / +8 | | / +8 |
%e | +7--5 |/ \|| |/ \| |/ \| |/ \|
%e 4------/ 3-------5 3-------5 3-------5 3-------5
%Y Cf. A180414, A338197, A338573, A338580, A338590, A338602.
%Y Cf. A338581, A338591, A338582, A338592 (similar for n = 11 and n = 12).
%K nonn,frac,fini,full
%O 1,1
%A _Hugo Pfoertner_, Nov 08 2020
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