A338487 a(n) is the number of non-isomorphic, serial/parallel indecomposable resistor networks with n edges, n >= 5, allowing dead ends.

1, 5, 36, 225, 1453, 9228, 58701, 372695, 2370155, 15117459, 96868355, 624326820, 4051597971, 26496771687, 174749567296, 1162909625384, 7812487626519, 53005074235282, 363305517314289, 2516343623698964, 17615995074375601, 124669825295709879, 892060223018406365

Offset: 5

Author

Rainer Rosenthal and Hugo Pfoertner, Oct 30 2020

Keywords

• nonn

Comments

A connected multigraph G with a selected pair P of nodes can be used to represent a resistor network. The edges represent resistors, and the total resistance is measured between the selected nodes. It is possible to construct complex networks using only serial or parallel combinations, but the more nodes and edges are involved, the more networks of a different kind can be found. They cannot be decomposed into serial/parallel elements. The sequence is on page 2 of the paper describing the computation of A180414 (see the Joel Karnofsky link).

Karnofsky claims that he systematically increased the number of edges by three basic operations, C, D, and E, defined in A338999, i.e., he claims to have counted the CDE-descendants of the simplest h-graph (the "bridge," see the example section). Numbers given in his paper are 1, 5, 37, 226, 1460, 9235, which is slightly off (see A339386). The difference seems to stem from the "dangling parts," as he calls them in his "addendum," so they don't affect the computation of different resistances in A180414. - Rainer Rosenthal, Dec 02 2020

Examples

			a(5) = 1. The only serial/parallel nondecomposable network with 5 resistors:
.
                      (+)-----A
     The "bridge"            / \
     see A337516            B---C
                             \ /
                      (-)-----Z
.
a(6) = 5. Constructed from the bridge with 5 resistors.
Allowed ways of adding a new edge are:
* an existing resistor is replaced by two parallel (N1, N2).
* a new resistor is appended (N3).
* an existing resistor is replaced by two serial (N4, N5).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
                    .                   .
         .-A        .         A         .         A
        / / \       .        / \        .   D    / \
       / /   \      .       /   \       .   |   /   \
      / /     \     .      /     \      .   |  /     \
     | /       \    .     /       \     .   | /       \
     |/         \   .    /.-------.\    .   |/         \
     B-----------C  .   B.         .C   .   B-----------C
      \         /   .    \`-------´/    .    \         /
       \       /    .     \       /     .     \       /
        \     /     .      \     /      .      \     /
         \   /      .       \   /       .       \   /
          \ /       .        \ /        .        \ /
           Z        .         Z         .         Z
                    .                   .
     N1: new edge   .   N2: new edge    .  N3: new node D
           A-B      .         B-C       .   with edge B-D
                    .                   .
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
                    .
           A        .         A
          / \       .        / \
         /   \      .       /   \
        D     \     .      /     \
       /       \    .     /       \
      /         \   .    /         \
     B-----------C  .   B-----D-----C
      \         /   .    \         /
       \       /    .     \       /
        \     /     .      \     /
         \   /      .       \   /
          \ /       .        \ /
           Z        .         Z
                    .
    N4: new node D  .  N5: new node D
     A-B now A-D-B  .   B-C now B-D-C
                    .
. . . . . . . . . . . . . . . . . . . . .
a(7) = 36. There are 24 interesting networks without dead ends.
See the pdf document with their description in the link section.
		

References

• Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Links

• Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
• Andrew Howroyd, PARI Program
• Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23, 2004.
• Rainer Rosenthal, Maple Program, Dec 02 2020.
• Rainer Rosenthal, The 24 networks with 7 resistors without dead ends (version 2), Feb 08 2021.

Crossrefs

Cf. A180414, A048211, A174283, A337516, A337517.

For graphs with two distinguished nodes see A304074.

Programs

Maple

SetA338487(5) := {"011111"}: # "bridge" adjacency matrix coded
for n from 6 to MAXEDGES do
   SetA338487(n) := C_D_E(SetA338487(n-1));  # see link section
od:
seq(nops(SetA338487(n)),n=1..MAXEDGES); # Rainer Rosenthal, Dec 02 2020

Extensions

a(10)-a(27) from Andrew Howroyd, Dec 02 2020