OFFSET
1,1
COMMENTS
The terms are sorted by increasing value of the resistance R(n) = a(n)/A338602(n).
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
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
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
LINKS
EXAMPLE
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.
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.
.
95/106 101/109 98/103 97/98 103/101
-1=======2 -1-------2 -1-------2 -1-------2 -1-------2
|\ /| |\ /|| |\ /| |\ /| |\ /|
| \ / | | \ / || | \ / | | \ / | | \ / |
| \ / | | \ / || | \ / | | \ / | | \ / |
| 4 | | 4 || | 4 | | 6 4 4 6 |
| / \ | | / \ || | //\ | | / \ | | / \ |
| / +6 | | / +6 || | // +6 | | / +7 | | / +7 |
|/ \| |/ \|| |// \| |/ \| |/ \|
3-------5 3-------5 3-------5 3-------5 3-------5
.
97/86 110/91 103/83 130/101 103/80
-1=======2 -1 -1-------2 -1-----2 -1=======2
| /| / \ | /|| | /|\ | /|
| / | / \ | 4 || | | | | | 4 |
| / | 2-----3 | / || | | | | | /| |
| 6 | ||\ / \ | 6 || | 4-6 | | / 6 |
| / \ | || \4/ | | / \ || | / | | | / | |
| 4 +7 | || \ | | / +7 || | / +7 | | / +7 |
|/ \| \\ +6--5 |/ \|| |/ \| |/ \|
3-------5 \\===// 3-------5 3-------5 3-------5
.
115/89 106/77 109/77 98/67 101/67
-1 -1-------2 -1-------2 -1-------2 -1-------2
/ \ | /|| | //| | /| | /|
/ \ | 4 || | 4 | | 6 | | 6 |
2-----3 | /| || | /| | | /| | | /| |
|\ / \ | / 6 || | / 6 | | / 7 | | / 7 4
| \6/ | | / | || | / | | | 4 | | | / | |
| \ | | / +7 || | / +7 | | / +8 | | / +8 |
| +7--5 |/ \|| |/ \| |/ \| |/ \|
4------/ 3-------5 3-------5 3-------5 3-------5
CROSSREFS
KEYWORD
nonn,frac,fini,full
AUTHOR
Hugo Pfoertner, Nov 08 2020
STATUS
approved