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A337517 a(n) is the number of distinct resistances that can be produced from a circuit with exactly n unit resistors. 22
1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3823, 11724, 36048, 110953, 342079, 1064468, 3341067, 10583564, 33727683 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

LINKS

Table of n, a(n) for n=0..18.

Wikipedia, Electrical resistance and conductance

Index to sequences related to resistances.

EXAMPLE

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

CROSSREFS

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

Sequence in context: A174283 A337516 A340920 * A268172 A151404 A027071

Adjacent sequences:  A337514 A337515 A337516 * A337518 A337519 A337520

KEYWORD

nonn,hard,more,nice

AUTHOR

Rainer Rosenthal and Hugo Pfoertner, Oct 29 2020

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

STATUS

approved

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)