A337517 a(n) is the number of distinct resistances that can be produced from a circuit with exactly n unit resistors.

1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3823, 11724, 36048, 110953, 342079, 1064468, 3341067, 10583564, 33727683, 107931482, 346615834

Offset: 0

Author

Rainer Rosenthal and Hugo Pfoertner, Oct 29 2020

Keywords

• nonn
• hard
• more
• nice

Comments

One can view a circuit with n unit resistors as a multigraph G with n edges and a pair P of distinguished nodes. Every edge of the graph must be contained in a path connecting the two distinguished nodes.

In case n > 0, a(n) counts all resistances R(G, P), which are rational numbers by Kirchhoff's laws. In case n = 0, the graph G consists of only two pair P nodes, and there is only one resistance: oo = infinity; so a(0) = 1. In the OEIS, there are already sequences that count the possible resistances of circuits of certain types (for the definitions see A337516).

OEIS | type | 1 2 3 4 5 6 7 8 9 10 11 12 13

---------+------+--------------------------------------------------------------

A048211 | SP | [1] 2 4 9 22 53 131 337 869 2213 5691 14517 37017

A174283 | SPB | 1 2 4 9 23 [57] 151 415 1157 3191 8687 23199 61677

A337516 | SPBF | 1 2 4 9 23 57 151 [421] 1202 3397 9498 25970 70005

A337517 | all | 1 2 4 9 23 57 151 [427] 1263 3823 11724 36048 110953

The table shows the number of different resistances, which grows with the complexity of the circuits. Values in square brackets mark the beginning of the newly explored range. Values a(n) up to n = 7 are fully classified, and have one of the given types, i.e., they can be computed by the functions Ser(), Par(), Bri(), and Frk() defined in A337516. For a(n), n >= 8, the theory in A180414 has to be applied.

Note: The 'set counted by A180414(n)' is the union of all 'sets counted by A337517(k) for k = 0 .. n'.

Admissible networks (G, P) are those defined in the Karnofsky paper (A180414).

Examples

			For a(n) up to n = 7 see the above mentioned sequences.
		

Links

• Wikipedia, Electrical resistance and conductance
• Index to sequences related to resistances.

Crossrefs

Cf. A048211, A180414, A174283, A337516, A338197.

Extensions

a(8)-a(14) from Andrew Howroyd, Oct 31 2020

a(15)-a(16) from Hugo Pfoertner, Dec 06 2020

a(17) from Hugo Pfoertner, Dec 09 2020

a(18) from Hugo Pfoertner, Apr 09 2021

a(19) from Zhao Hui Du, May 15 2023

a(20) from Zhao Hui Du, May 23 2023