A180414 -id:A180414 - cp's OEIS frontend

A338573 Array read by ascending antidiagonals: T(m,n) (m, n >= 1) is the minimum number of unit resistors needed to produce resistance m/n.

1, 2, 2, 3, 1, 3, 4, 3, 3, 4, 5, 2, 1, 2, 5, 6, 4, 4, 4, 4, 6, 7, 3, 4, 1, 4, 3, 7, 8, 5, 2, 5, 5, 2, 5, 8, 9, 4, 5, 3, 1, 3, 5, 4, 9, 10, 6, 5, 5, 5, 5, 5, 5, 6, 10, 11, 5, 3, 2, 5, 1, 5, 2, 3, 5, 11, 12, 7, 6, 6, 5, 5, 5, 5, 6, 6, 7, 12, 13, 6, 6, 4, 6, 4, 1, 4, 6, 4, 6, 6, 13

Offset: 1

Rainer Rosenthal, Nov 05 2020

Karnofsky (2004, p. 5): "[...] if some circuit has resistance m/n then some other circuit likely has n/m. In fact, for 9 or fewer resistors, this symmetry is perfect. However, for 10 resistors the following values are achieved, but not their inverses: 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". That means, that T(m,n) = T(n,m), if T(m,n) <= 9.
This starts with the values of A113881, but the Karnofsky comment says that T(n,m) is not symmetric, whereas the count of tiles in A113881 is. - R. J. Mathar, Nov 06 2020
The first difference where T(m,n) = T(n,m), but differs from the corresponding entry of A113881 occurs for (n,m) = (154,167) and (n,m) = (167,154), both representable by networks with non-planar graphs of 11 resistors, whereas A113881 counts 12 tiles. See Pfoertner link for illustration of more differences. - Hugo Pfoertner, Nov 13 2020

			T(1,2) = 2: at least 2 unit resistors in parallel are needed for resistance 1/2.
T(2,1) = 2: at least 2 unit resistors in series are needed for resistance 2 = 2/1.
T(11,13) = 6: the following "bridge" has resistance Bri(Par(1,1),1,1,1,1) = 11/13 (see A337516 for definitions):
.
                  (+)
                  / \
              ---*   \
             /  /     \
           (1)(1)     (1)
             \ |       |
              \|       |
               *--(1)--*
                \     /
                (1) (1)
                  \ /
                  (-)
.
T(13,11) = 6: Bri(Ser(1,1),1,1,1,1) = 13/11.
T(95,106) = 10, but T(106,95) > 10: Karnofsky (2004, p. 5), see comment.

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

Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.
Hugo Pfoertner, Where A338573 differs from A113881, x,y <= 380.
Index to sequences related to resistances.

Cf. A048211, A113881, A180414, A174283, A337516, A337517.

Non-reciprocal ratios: A338601/A338602 (10 resistors), A338581/A338591 (11 resistors), A338582/A338592 (12 resistors).

A338602 Denominators y 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. Numerators are in A338601.

Original entry on oeis.org

106, 109, 103, 98, 101, 86, 91, 83, 101, 80, 89, 77, 77, 67, 67

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of R.

			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.

Index to sequences related to resistances.

Cf. A180414, A338601.

A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.

Original entry on oeis.org

1, 2, 4, 8, 20, 42, 102, 250, 610, 1486, 3710, 9228, 23050, 57718, 145288, 365820, 922194, 2327914, 5885800, 14890796, 37701452, 95550472, 242325118, 614869792, 1561228066, 3966071764, 10080113232, 25630109268, 65194419268, 165890640468

Offset: 1

Hugo van der Sanden

If x and y require xn and yn resistors respectively, then (x+y) and 1/(1/x + 1/y) require no more than (xn+yn). Inspired by a sci.math posting by Miguel A. Lerma (lerma(AT)math.nwu.edu).
Let A(n) be the set of resistances equivalent to a network of n 1-ohm resistors using only series and parallel combinations. Then A048211(n) = card(A(n)). Let L(n) be the set of resistances that first appear in A(n), i.e. L(n) = A(n) \ (A(1) U ... U A(n-1)). Then a(n) = card(L(n)). - Antoine Mathys, Nov 22 2024
If a resistance is equivalent to a n-resistor circuit, then it is equivalent to a 4n-resistor circuit. There is therefore no upper bound on the size of the networks to which it is equivalent. - Antoine Mathys, Nov 22 2024

			The a(1) = 1 resistance value is 1 ohm.
The a(2) = 2 resistance values are {1/2, 2}.
The a(3) = 4 resistance values are {1/3, 2/3, 3/2, 3}.
The a(4) = 8 resistance values are {1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4}.
The a(5) = 20 resistance values are {1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/6, 6/7, 7/6, 6/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, 5}.
E.g. 6/5 is made from two resistors in series in parallel with three resistors in series, since 6/5 = 1/(1/2 + 1/3). It cannot be obtained using fewer resistors.

Miguel A. Lerma, resistors, post in the newsgroup sci.math, Nov 5 1999.
Index to sequences related to resistances.

Cf. A048211, A046825, A153588, A180414, A338197.

a(n) = A153588(n) - A153588(n-1) for n > 1. - Hugo Pfoertner, Nov 04 2020

a(15)-a(21) from Jon E. Schoenfield, Aug 28 2006
Definition corrected by Jon E. Schoenfield, Aug 27 2006
a(22)-a(23) from Graeme McRae, Aug 18 2007
a(24)-a(25) from Antoine Mathys, Mar 20 2017
Definition changed to say "exactly". - N. J. A. Sloane, Nov 07 2020
Definition clarified by Antoine Mathys, Nov 22 2024
a(26)-a(30) from Antoine Mathys, Dec 05 2024

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

Original entry on oeis.org

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

Rainer Rosenthal and Hugo Pfoertner, Oct 30 2020

nonn

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

			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.

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

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.

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

For graphs with two distinguished nodes see A304074.

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

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

A338581 Numerators x of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338591.

Original entry on oeis.org

95, 101, 98, 97, 103, 97, 110, 103, 130, 103, 115, 106, 109, 119, 106, 98, 116, 109, 101, 124, 121, 111, 121, 136, 124, 151, 141, 169, 121, 151, 134, 136, 125, 133, 127, 134, 136, 149, 142, 146, 161, 137, 161, 142, 146, 145, 152, 149, 169, 161, 151, 167, 175, 149, 151, 194, 176, 150, 166, 174

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of R.

See A338601 for more information.

Hugo Pfoertner, Table of n, a(n) for n = 1..172
Hugo Pfoertner, Visualization of the non-reciprocal resistance values with 11 resistors, using Plot 2.

Cf. A180414, A338591, A338592, A338601, A338602.

First differs from A338582 for n=59.

A338591 Denominators y of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338581.

Original entry on oeis.org

201, 210, 201, 195, 204, 183, 201, 186, 231, 183, 204, 183, 186, 201, 179, 165, 195, 183, 168, 191, 186, 169, 183, 199, 175, 209, 194, 231, 164, 204, 181, 183, 168, 178, 169, 173, 175, 191, 179, 181, 199, 167, 194, 169, 173, 166, 167, 162, 183, 174, 161, 178, 186

Offset: 1

Hugo Pfoertner, Nov 08 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..172

Cf. A180414, A338591, A338601, A338602.

A338607 Resistance values R < 1 ohm, multiplied by a common denominator 232792560 (= A338600(7)), that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors.

Original entry on oeis.org

33256080, 42325920, 49884120, 53721360, 62078016, 64664600, 68468400, 71628480, 72747675, 73513440, 81477396, 82162080, 83140200, 85765680, 88682880, 90530440, 95855760, 98017920, 101846745, 106696590, 110270160, 110853600, 121938960, 122522400, 126095970

Offset: 1

Hugo Pfoertner, Nov 05 2020

The list of the A338197(7)/2 = 57 resistance values < 1 ohm is A338587(n)/A338597(n). a(n) = 232792560 * [1/7, 2/11, 3/14, 3/13, 4/15, 5/18, 5/17, ..., 19/21, 11/12, 12/13, 13/14, 14/15, 15/16, 18/19].

Hugo Pfoertner, Table of n, a(n) for n = 1..57

Cf. A180414, A338197, A338587, A338597, A338600, A338605, A338606, A338608, A338609.

A338582 Numerators x of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338592.

Original entry on oeis.org

95, 101, 98, 97, 103, 97, 110, 103, 130, 103, 115, 106, 109, 119, 106, 98, 116, 109, 101, 124, 121, 111, 121, 136, 124, 151, 141, 169, 121, 151, 134, 136, 125, 133, 127, 134, 136, 149, 142, 146, 161, 137, 161, 142, 146, 145, 152, 149, 169, 161, 151, 167, 175, 149, 151, 194, 176, 150, 152, 166

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of R.

See A338601 for more information.

Hugo Pfoertner, Table of n, a(n) for n = 1..1114
Hugo Pfoertner, Visualization of the non-reciprocal resistance values with 12 resistors, using Plot 2.
Hugo Pfoertner, Comparison of numerators for 11 and 12 resistors, using Plot 2.

Cf. A180414, A338581, A338591, A338592, A338601, A338602.

A338592 Denominators y of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338582.

Original entry on oeis.org

296, 311, 299, 292, 307, 280, 311, 289, 361, 286, 319, 289, 295, 320, 285, 263, 311, 292, 269, 315, 307, 280, 304, 335, 299, 360, 335, 400, 285, 355, 315, 319, 293, 311, 296, 307, 311, 340, 321, 327, 360, 304, 355, 311, 319, 311, 319, 311, 352, 335, 312, 345, 361

Offset: 1

Hugo Pfoertner, Nov 08 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..1114

Cf. A180414, A338581, A338582, A338591, A338601, A338602.

A338608 Resistance values R < 1 ohm, multiplied by a common denominator 5342931457063200 (= A338600(8)), that can be obtained from a network of exactly 8 one-ohm resistors, but not from any network with fewer than 8 one-ohm resistors.

Original entry on oeis.org

667866432132900, 821989454932800, 942870257128800, 1001799648199350, 1124827675171200, 1161506838492000, 1214302603878000, 1257160342838400, 1272126537396000, 1282303549695168, 1385204451831200, 1393808206190400, 1406034593964000, 1438481546132400, 1473912126086400

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the A338197(8)/2 = 156 resistance values < 1 ohm is A338580(n)/A338598(n). a(n) = 5342931457063200 * [1/8, 2/13, 3/17, 3/16, 4/19, 5/23, 5/22, ..., 23/24, 24/25, 25/26, 26/27, 27/28, 30/31, 34/35].

Hugo Pfoertner, Table of n, a(n) for n = 1..156

Cf. A180414, A338197, A338580, A338598, A338600.

Cf. A338605, A338606, A338607, A338609 (similar for n = 5..9).

cp's OEIS Frontend

A338573 Array read by ascending antidiagonals: T(m,n) (m, n >= 1) is the minimum number of unit resistors needed to produce resistance m/n.

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A338602 Denominators y 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. Numerators are in A338601.

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A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.

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

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A338581 Numerators x of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338591.

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A338591 Denominators y of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338581.

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A338607 Resistance values R < 1 ohm, multiplied by a common denominator 232792560 (= A338600(7)), that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors.

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A338582 Numerators x of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338592.

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A338592 Denominators y of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338582.

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A338608 Resistance values R < 1 ohm, multiplied by a common denominator 5342931457063200 (= A338600(8)), that can be obtained from a network of exactly 8 one-ohm resistors, but not from any network with fewer than 8 one-ohm resistors.

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