A338573 -id:A338573 - cp's OEIS frontend

A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

1, 2, 4, 8, 16, 36, 80, 194, 506, 1400, 4039, 12044, 36406, 111324, 342447, 1064835, 3341434, 10583931, 33728050, 107931849, 346616201

Offset: 0

Vaclav Kotesovec, Sep 02 2010

In "addendum" J. Karnofsky stated the value a(15) = 1064833. In contrast to the terms up to and including a(14), which could all be confirmed, an independent calculation based on a list of 3-connected simple graphs resulted in the corrected value a(15) = 1064835. - Hugo Pfoertner, Dec 06 2020

See A337517 for the number of different resistances that can be obtained by combining /exactly/ n one-ohm resistors. The method used by Andrew Howroyd (see his program in the link section) uses 3-connected graphs with one edge (the 'battery edge') removed. - Rainer Rosenthal, Feb 07 2021

			a(n) counts all resistances that can be obtained with fewer than n resistors as well as with exactly n resistors. Without a resistor the resistance is infinite, i.e., a(0) = 1. One 1-ohm resistor adds resistance 1, so a(1) = 2. Two resistors in parallel give 1/2 ohm, while in series they give 2 ohms. So a(2) is the number of elements in the set {infinity, 1, 1/2, 2}, i.e., a(2) = 4. - _Rainer Rosenthal_, Feb 07 2021

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 (includes scripts for other related sequences)
Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.
Joel Karnofsky, Mathematica program for resistances, copied from article.
Index to sequences related to resistances.

Cf. A048211, A051389, A174284, A337517, A338197, A338487, A338573, A338580, A338590, A338600.

Cf. A123545, A338511, A338999, A339072.

Mathematica
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(* See link. *)
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a(n) = A174284(n) + 1 for n <= 7, a(n) > A174284(n) + 1 otherwise. - Hugo Pfoertner, Nov 01 2020

a(n) is the number of elements in the union of the sets SetA337517(k), k <= n, counted by A337517. - Rainer Rosenthal, Feb 07 2021

a(15) corrected and a(16) added by Hugo Pfoertner, Dec 06 2020
a(17) from Hugo Pfoertner, Dec 09 2020
a(0) from Rainer Rosenthal, Feb 07 2021
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

A113881 Table of smallest number of squares, T(m,n), needed to tile an m X n rectangle, read by antidiagonals.

Original entry on oeis.org

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, 14, 8, 4, 6, 2, 3, 7, 7, 3, 2, 6, 4, 8, 14

Offset: 1

Devin Kilminster (devin(AT)27720.net), Jan 27 2006

a(n) = A338573(n) for n <= 105, as stated by R. J. Mathar. These sequences are essentially different though, because a(13433) = T(67,98) = T(98,67) = a(13464), but A338573(13433) != A338573(13464). The relationship between the tiling problem and resistor networks is remarkable. There are explanations in M. Ortolano et al., 2013. - Rainer Rosenthal, Nov 09 2020

			T(n,n) = 1 (1 n X n square).
T(n,1) = n (n 1 X 1 squares).
T(6,7) = 6 (2 3 X 3, 1 4 X 4, 1 2 X 2, 2 1 X 1).
T(11,13) = 6 (1 7 X 7, 1 6 X 6, 1 5 X 5, 2 4 X 4 1 1 X 1).
Table T(m,n) begins:
:   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
:   2, 1, 3, 2, 4, 3, 5, 4, 6,  5, ...
:   3, 3, 1, 4, 4, 2, 5, 5, 3,  6, ...
:   4, 2, 4, 1, 5, 3, 5, 2, 6,  4, ...
:   5, 4, 4, 5, 1, 5, 5, 5, 6,  2, ...
:   6, 3, 2, 3, 5, 1, 5, 4, 3,  4, ...
:   7, 5, 5, 5, 5, 5, 1, 7, 6,  6, ...
:   8, 4, 5, 2, 5, 4, 7, 1, 7,  5, ...
:   9, 6, 3, 6, 6, 3, 6, 7, 1,  6, ...
:  10, 5, 6, 4, 2, 4, 6, 5, 6,  1, ...

Alois P. Heinz, Antidiagonals n = 1..350, flattened (using data from A219158)
Bertram Felgenhauer, Filling rectangles with integer-sided squares
Richard J. Kenyon, Tiling a rectangle with the fewest squares, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.
M. Ortolano, M. Abrate, and L. Callegaro, On the synthesis of Quantum Hall Array Resistance Standards, arXiv preprint arXiv:1311.0756 [physics.ins-det], 2013.
Mark Walters, Rectangles as sums of squares, Discrete Math. 309 (2009), no. 9, 2913-2921.

Rows (or columns) m=1-10 give: A001477, A030451, A226576, A226577, A226578, A226579, A226580, A226581, A226582, A226583.

Cf. A219158, A219924, A226545, A338573.

(* *** Warning *** This empirical toy-program is based on the greedy algorithm. Its output was only verified for n+k <= 32. Any use outside this domain might produce only upper bounds instead of minimums. *)
nmax = 31; Clear[T];
Tmin[n_, k_] := Table[{1 + T[ c, k - c] + T[n - c, k], 1 + T[n, k - c] + T[n - c, c]}, {c, 1, k - 1}] // Flatten // Min;
Tmin2[n_, k_] := Module[{n1, n2, k1, k2}, 1 + T[n2, k1 + 1] + T[n - n1, k2] + T[n - n2, k1] + T[n1, k - k1] /. {Reduce[1 <= n1 <= n - 1 && 1 <= n2 <= n - 1 && 1 <= k1 <= k - 1 && 1 <= k2 <= k - 1 && n1 + 1 + n2 == n && k1 + 1 + k2 == k, Integers] // ToRules} // Min];
T[n_, n_] = 1;
T[n_, 1] := n;
T[1, k_] := k;
T[n_, k_ /; k > 1] /; n > k && Divisible[n, k] := n/k;
T[n_, k_ /; k > 1] /; n > k := T[n, k] = If[k >= 5 && n >= 6 && n - k <= 3, Min[Tmin[n, k], Tmin2[n, k], T[k, n - k] + 1], T[k, n - k] + 1];
T[n_, k_ /; k > 1] /; n < k := T[n, k] = T[k, n];
Table[T[n - k + 1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 11 2016, checked against first 496 terms of the b-file *)

A338601 Numerators x 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. Denominators are in A338602.

Original entry on oeis.org

95, 101, 98, 97, 103, 97, 110, 103, 130, 103, 115, 106, 109, 98, 101

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of the resistance R(n) = a(n)/A338602(n).
For more information, references, and links, see A180414 and A338573.
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

			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

Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
Rainer Rosenthal, Karnofsky's 15 exceptional networks with 10 resistors (Version 2).
Index to sequences related to resistances.

Cf. A180414, A338197, A338573, A338580, A338590, A338602.

Cf. A338581, A338591, A338582, A338592 (similar for n = 11 and n = 12).

A338590 Denominators of resistance values that can be obtained from a network of exactly 10 one-ohm resistors, but not from any network with fewer than 10 one-ohm resistors. Numerators are in A338580.

Original entry on oeis.org

10, 17, 23, 22, 27, 33, 32, 25, 31, 37, 41, 35, 29, 40, 45, 39, 38, 43, 37, 57, 46, 51, 54, 49, 63, 43, 52, 47, 51, 60, 55, 41, 50, 59, 58, 49, 40, 53, 57, 48, 61, 43, 47, 64, 38, 59, 67, 71, 29, 33, 78, 45, 49, 53, 57, 61, 73, 75, 63, 59, 55, 51, 47, 82, 35, 31

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the 2639 = A338197(10) resistance values, sorted by increasing size of R = A338580(n)/a(n), is [1/10, 2/17, 3/23, 3/22, 4/27, 5/33, 5/32, ..., 32/5, 33/5, 27/4, 22/3, 23/3, 17/2, 10]. There are 15 terms for which their reciprocal value is not in the sequence, given in A338601/A338602.

Hugo Pfoertner, Table of n, a(n) for n = 1..2639

Cf. A180414, A338197, A338573, A338580, A338600, A338601, A338602.

Cf. A338595, A338596, A338597, A338599, A338599 (similar for n = 5..10).

Title corrected by Rainer Rosenthal, Feb 14 2021

A338598 Denominators of resistance values < 1 ohm 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. Numerators are in A338580.

Original entry on oeis.org

8, 13, 17, 16, 19, 23, 22, 17, 21, 25, 27, 23, 19, 26, 29, 25, 24, 27, 23, 35, 28, 31, 32, 29, 37, 25, 30, 27, 29, 34, 31, 23, 28, 33, 32, 27, 22, 29, 31, 26, 33, 23, 25, 34, 20, 31, 35, 37, 15, 17, 40, 23, 25, 27, 29, 31, 37, 37, 31, 29, 27, 25, 23, 40, 17, 15

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the 312 = A338197(8) resistance values, sorted by increasing size of R = A338580(n)/a(n) = A338608(n)/A338600(8), is the union of [1/8, 2/13, 3/17, ..., 27/28, 30/31, 34/35] and of the corresponding reciprocal resistances > 1 ohm [35/34, 31/30, 28/27, ..., 17/3, 13/2, 8].

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

Cf. A338197, A338573, A338580, A338600, A338608.

Cf. A338595, A338596, A338597, A338599, A338590 (similar for n = 5..10).

A338583 Number of unlabeled 3-connected nonplanar graphs with n edges.

Original entry on oeis.org

1, 2, 3, 10, 29, 94, 343, 1291, 5206, 22061, 96908, 439837, 2053916, 9841412, 48319944, 242857491, 1248629027, 6563581656, 35258560001, 193463945790

Offset: 9

Hugo Pfoertner, Nov 21 2020

Hugo Pfoertner, Illustrations of terms a(9) - a(12).

Cf. A002840, A006290, A337517, A338197, A338511, A338573, A338584, A338593, A338594, A338604.

a(n) = A338511(n) - A002840(n).

a(n) <= A338593(n). The difference A338584(n) = A338593(n)-a(n) are the counts of nonplanar connected graphs with minimum degree 3 at each node that are not 3-connected.

A348020 a(n) is the minimum number of unit resistors in a circuit with resistance R = A007305(n)/A007306(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Hugo Pfoertner, Oct 14 2021

nonn

For small values of n, the circuits are planar and correspond to the tiling of rectangles by squares. See A338573 for more information and examples.

The earliest nonplanar deviation occurs at a(3173) corresponding to R = 115/204 needing 11 instead of 12 resistors.

Hugo Pfoertner, Table of n, a(n) for n = 1..4096

First 31 terms coincide with A070941.
Cf. A007305, A007306, A113881, A219158, A338573, A347282.
A338579 can be used for a lookup of the position for a given rational R.

A339547 a(n) is the number of resistance values R=x/y that can be obtained by a network of at most n one-ohm resistors such that a network of more than n one-ohm resistors is needed to obtain the resistance y/x.

Original entry on oeis.org

15, 172, 1114, 5378, 22321, 83995, 293744, 968965

Offset: 10

Hugo Pfoertner, Dec 10 2020

a(n) = 0 for n < 10.

			a(10) = 15: this is the number of non-reciprocal resistance values provided in Karnofsky's solution of the 10-resistors puzzle. The list of 15 resistances is: 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.
a(11) = 172: the corresponding resistances are provided in A338581/A338591.
a(12) = 1114: the corresponding resistances are provided in A338582/A338592.

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.

Cf. A180414, A338573, A338601, A338602, A338581, A338591, A338582, A338592.

A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 24, 40, 69, 130, 231, 408, 689, 1272, 2153, 3960, 6993, 12560

Offset: 1

Rainer Rosenthal, Jan 16 2021

a(n) is taken from the set of resistance values counted by A337517(n). These sets can be computed by the PARI program of Andrew Howroyd in A180414.
Also the maximum numerator of these electrical networks for small n.
Maximum numerator and maximum denominator coincide for planar networks: for every resistance R in a planar network with n resistors there is always another planar network with n resistors and resistance 1/R. For nonplanar networks this is not necessarily so, as can be seen in A338573.
The asymmetry is illustrated by the example a(15) = 2153.
The author conjectures that this asymmetry will increase with n, and eventually the maxima will differ.
Conjecture: a(19) = 22233, a(20) = 39918. It would be very desirable to know at which value of n > 18 the maximum values of numerators and denominators differ for the first time. - Hugo Pfoertner, Apr 19 2021

			Denominators for numerator a(15) = 2153 in electrical networks with 15 resistors:
  1025,1049,1051,1058,1089,1104,1145,1184,1185,1193,1208,
  1212,1219,1248,1254,1337,1382,1403,1526,1527,1529,1530,
  1545,1547,1555,1579,1586,1632,1642,1647,1687,1699,1719.
Numerators for denominator a(15) = 2153 in electrical networks with 15 resistors:
   899, 905, 934, 941, 945, 960, 968, 969,1008,1049,1064,
  1095,1102,1104,1128,1137,1143,1147,1164,1182,1207,1296,
  1359,1367,1387,1400,1447,1543.

Index to sequences related to resistances.

Cf. A180414, A337517, A338601, A339548, A339808, A340726.

a(18) from Hugo Pfoertner, Apr 11 2021

cp's OEIS Frontend

A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

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A113881 Table of smallest number of squares, T(m,n), needed to tile an m X n rectangle, read by antidiagonals.

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A338601 Numerators x 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. Denominators are in A338602.

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A338590 Denominators of resistance values that can be obtained from a network of exactly 10 one-ohm resistors, but not from any network with fewer than 10 one-ohm resistors. Numerators are in A338580.

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A338598 Denominators of resistance values < 1 ohm 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. Numerators are in A338580.

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A338583 Number of unlabeled 3-connected nonplanar graphs with n edges.

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A348020 a(n) is the minimum number of unit resistors in a circuit with resistance R = A007305(n)/A007306(n).

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A339547 a(n) is the number of resistance values R=x/y that can be obtained by a network of at most n one-ohm resistors such that a network of more than n one-ohm resistors is needed to obtain the resistance y/x.

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A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

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