A337517 -id:A337517 - 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
```
(* See link. *)
```

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

A174283 Number of distinct resistances that can be produced using n equal resistors in, series, parallel and/or bridge configurations.

Original entry on oeis.org

1, 2, 4, 9, 23, 57, 151, 415, 1157, 3191, 8687, 23199, 61677, 163257, 432541, 1146671, 3039829

Offset: 1

Sameen Ahmed Khan, Mar 15 2010

This sequence is a variation on A048211, which uses only series and parallel combinations. Since a bridge circuit requires minimum of five resistances the first four terms coincide. For the definition of "bridge" see A337516.

			Example 1: Five unit resistors: each arm of the bridge has one unit resistor, leading to an equivalent resistance of 1; so the set is {1} and its order is 1. Thus a(5) = A048211(5) + 1 = 23.
Example 2: Six unit resistors: a bridge with 6 resistors yields A174285(6) = 3 different resistances and the series parallel combinations give A048211(6) = 53 resistances, but resistance 1 is counted twice. The union of the forementioned resistances has cardinality 53+3-1 = 55. There are two more circuits to be considered: the bridge with five unit resistors and the sixth unit resistor either in parallel (value 1/2) or in series (value 2). Both values 1/2 and 2 are not counted by A048211(6) or A174285(6), so the total is 55 + 2 = 57. - _Rainer Rosenthal_, Oct 25 2020

Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162.
Sameen Ahmed Khan, Beginning to Count the Number of Equivalent Resistances, Indian Journal of Science and Technology, Vol. 9, Issue 44, pp. 1-7, 2016.
Hugo Pfoertner, Increase of number of representable resistances by allowing bridges, Plot2 of a(n)/A048211.

Cf. A048211, A153588, A174284, A174285, A174286, A176499, A176500, A176501, A176502, A180414, A337516, A337517.

a(8) corrected and a(9)-a(17) from Rainer Rosenthal, Oct 29 2020

A338197 a(n) is the number of distinct resistances that can be obtained by a network of exactly n equal resistors, but not by any network with fewer than n equal resistors.

Original entry on oeis.org

1, 2, 4, 8, 20, 44, 114, 312, 894, 2639, 8005, 24362, 74918, 231123, 722388, 2276599, 7242497, 23144119, 74203799, 238684352

Offset: 1

Hugo Pfoertner, Nov 03 2020

See A180414 and A337517 for more information and references.

			a(6) = 44 because the resistances 11/13 and 13/11 (in units of resistor value) are representable in addition to the A051389(6)=42 resistances that can be achieved by only serial and parallel configurations with exactly 6 resistors and not by a network with fewer than 6 resistors.

Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
Index to sequences related to resistances.

Cf. A051389, A180414, A337517, A338487.

a(n) = A180414(n) - A180414(n-1).

a(n) = A051389(n) for n <= 5, a(n) > A051389(n) otherwise.

a(15) corrected and a(16) added by 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

A174286 Number of distinct resistances that can be produced using at most n equal resistors in series and/or parallel, confined to the five arms (four arms and the diagonal) of a bridge configuration. Since the bridge requires a minimum of five resistors, the first four terms are zero.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 19, 75, 291, 985, 3011, 8659, 24319, 65899, 176591, 464451, 1211185

Offset: 1

Sameen Ahmed Khan, Mar 15 2010

			Example 1: Five equal unit resistors. Each arm of the bridge has one unit resistor, leading to an equivalent resistance of 1; so the set is {1} and its order is 1. Example 2: Six equal unit resistors. Four arms have one unit resistor each and the fifth arm has two unit resistors. Two resistors in the same arm, when combined in series and parallel result in 2 and 1/2 respectively (corresponding to 2: {1/2, 2} in A048211). The set {1/2, 2}, in the diagonal results in {1}. Set {1/2, 2} in any of the four arms results in {11/13, 13/11}. Consequently, with six equal resistors, we have the set {11/13, 1, 13/11}, whose order is 3. Union of the previous terms is {1} and the union with these three is again {11/13, 1, 13/11}. So the terms for five and six resistors are 1 and 3 respectively.

Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). Digital Object Identifier (DOI): 10.1119/1.19396.
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
Rainer Rosenthal, Maple programs SetA174285 and SetA174286
Marx Stampfli, Bridged graphs, circuits and Fibonacci numbers, Applied Mathematics and Computation, Volume 302, 1 June 2017, Pages 68-79.
Index to sequences related to resistances.

Cf. A048211, A153588, A174283, A174284, A174285, A180414, A337516, A337517, A338487.

Cf. A176499, A176500, A176501, A176502.

See link section: A174286(n) = nops(SetA174286(n)).

From Stampfli's paper, a(8) corrected and a(9)-a(12) added by Eric M. Schmidt, Sep 09 2017
Name edited by Eric M. Schmidt, Sep 09 2017
a(13)-a(17) added by Rainer Rosenthal, Feb 05 2021

A176502 a(n) = 2*Farey(m; I) - 1 where m = Fibonacci (n + 1) and I = [1/n, 1].

Original entry on oeis.org

1, 3, 7, 17, 37, 99, 243, 633, 1673, 4425, 11515, 30471, 80055, 210157, 553253, 1454817, 3821369, 10040187, 26360759, 69201479, 181628861, 476576959, 1250223373, 3279352967, 8600367843, 22551873573, 59128994931, 155014246263, 406350098913, 1065104999651

Offset: 1

Sameen Ahmed Khan, Apr 21 2010

nonn

This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. This sequence provides a better strict upper bound than A176500 but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 10 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 25 2021

			n = 5, , I = [1/5, 1], m = Fibonacci(6) = 8, Farey(8) = 23, Farey(8; I) = 19, Grand Set(5) = 37.

Antoine Mathys, Table of n, a(n) for n = 1..40
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
Sameen Ahmed Khan, Integer Sequences Authored by Dr. Sameen Ahmed Khan
Sameen Ahmed Khan, Mathematica notebook
Sameen Ahmed Khan, How Many Equivalent Resistances?, RESONANCE, May 2012. - From _N. J. A. Sloane_, Oct 15 2012
Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - From _N. J. A. Sloane_, Oct 23 2012
Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).
Hugo Pfoertner, Ratio for planar networks with generalized bridges, Plot2 of A337516(n)/a(n).
Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176501, A337516, A337517.

a1[n_ /; n<4] := 2^(n-1); a1[n_] := Module[{m = Fibonacci[n+1], v}, v = Reap[Do[Sow[j/i], {i, n+1, m}, {j, 1, (i-1)/n}]][[2, 1]]; Total[EulerPhi[ Range[m]]] - Length[v // Union]];
a[n_] := 2 a1[n] - 1;
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* Jean-François Alcover, Aug 30 2018, after Antoine Mathys *)

farey(n) = sum(i=1, n, eulerphi(i)) + 1;
a176501(n) = my(m=fibonacci(n + 1), count=0); for(b=n+1, m, for(a=1, (b-1)/n, if(gcd(a,b)==1, count++))); farey(m) - 1 - count;
a(n) = 2 * a176501(n) - 1; \\ Antoine Mathys, May 07 2019

a(n) = 2 * A176501(n) - 1. - Antoine Mathys, Aug 07 2018

a(19)-a(27) from Antoine Mathys, Aug 10 2018

a(28)-a(30) from Antoine Mathys, May 07 2019

A338600 a(n) is the common denominator of the A338197(n) rational resistance values that can be obtained from a network of exactly n one-ohm resistors, but not by a network of fewer than n one-ohm resistors.

Original entry on oeis.org

1, 2, 6, 60, 840, 360360, 232792560, 5342931457063200, 591133442051411133755680800, 79057815923102180093748328364591874435251553600

Offset: 1

Hugo Pfoertner, Nov 03 2020

nonn

The next terms a(11)=8.87124454467...*10^84 and a(12)=1.80685581583...*10^141 are too big to be included in the data.

			a(4) = 60: The resistance values for which a minimum of 4 resistors is needed are [1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4] with a common denominator of 60.
a(1) = 1: [1],
a(2) = 2: [1/2, 2],
a(3) = 6: [1/3, 2/3, 3/2, 3].

Hugo Pfoertner, Table of n, a(n) for n = 1..15

Cf. A048211, A051389, A180414, A337517, A338197, A338580, A338590, A338595, A338596, A338597, A338598, A338599, A338605, A338606, A338607, A338608, A338609.

A176499 Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).

Original entry on oeis.org

2, 3, 5, 11, 23, 59, 141, 361, 941, 2457, 6331, 16619, 43359, 113159, 296385, 775897, 2030103, 5315385, 13912615, 36421835, 95355147, 249635525, 653525857, 1710966825, 4479358275, 11726974249, 30701593527, 80377757397, 210431301141, 550916379293

Offset: 1

Sameen Ahmed Khan, Apr 21 2010

nonn

This sequence arises in the analytically obtained strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides a strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176501 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]
Farey(n) = A005728(n). [Franklin T. Adams-Watters, May 12 2010]
The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is wrong. It only applies to purely serial-parallel networks (A048211), but it already fails when bridges are allowed, as described in A174283. Even more so if arbitrary nonplanar networks are allowed as in A337517. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 24 2021

			n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23.

Antoine Mathys, Table of n, a(n) for n = 1..50
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). Digital Object Identifier (DOI): 10.1119/1.19396.
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
Sameen Ahmed KHAN, Mathematica notebook 1
Sameen Ahmed KHAN, Mathematica notebook 2
Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).
Hugo Pfoertner, Ratio for networks with bridges, Plot2 of A174283(n)/a(n).
Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).

Cf. A000045, A005728.

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176500, A176501, A176502, A337517.

List([1..30],n->Sum([1..Fibonacci(n+1)],i->Phi(i)))+1; # Muniru A Asiru, Jul 31 2018

[1+&+[EulerPhi(i):i in [1..Fibonacci(n+1)]]:n in [1..30]]; // Marius A. Burtea, Jul 26 2019

with(numtheory): with(combinat,fibonacci): a:=n->1+add(phi(i),i=1..n): seq(a(fibonacci(n+1)),n=1..30); # Muniru A Asiru, Jul 31 2018

b[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
a[n_] := b[Fibonacci[n + 1]];
Array[a, 30] (* Jean-François Alcover, Sep 20 2018 *)

farey(n) = 1+sum(k=1, n, eulerphi(k));
a(n) = farey(fibonacci(n+1)); \\ Michel Marcus, Jul 31 2018

a(n) = A005728(A000045(n+1)). - Michel Marcus, Jul 31 2018

a(26)-a(29) from Sameen Ahmed Khan, May 02 2010

a(30) from Antoine Mathys, Aug 06 2018

A176500 a(n) = 2*Farey(Fibonacci(n + 1)) - 3.

Original entry on oeis.org

1, 3, 7, 19, 43, 115, 279, 719, 1879, 4911, 12659, 33235, 86715, 226315, 592767, 1551791, 4060203, 10630767, 27825227, 72843667, 190710291, 499271047, 1307051711, 3421933647, 8958716547, 23453948495, 61403187051, 160755514791, 420862602279, 1101832758583

Offset: 1

Sameen Ahmed Khan, Apr 21 2010

nonn

This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176502 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, Jul 12 2019]
Farey(n) = A005728(n). - Franklin T. Adams-Watters, May 12 2010
The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 11 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 24 2021

			n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23, 2Farey(m) - 3 = 43.

Antoine Mathys, Table of n, a(n) for n = 1..50
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (Apr 20 2010).
Sameen Ahmed KHAN, Mathematica notebook 1
Sameen Ahmed KHAN, Mathematica notebook 2
Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).
Hugo Pfoertner, Ratio for planar networks with generalized bridges, Plot2 of A337516(n)/a(n).
Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176501, A176502.

[2*(&+[EulerPhi(k):k in [1..Fibonacci(n+1)]])-1:n in [1..30]]; // Marius A. Burtea, Jul 26 2019

a[n_] := 2 Sum[EulerPhi[k], {k, 1, Fibonacci[n+1]}] - 1;
Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* Jean-François Alcover, Nov 03 2018, from PARI *)

a(n) = 2*sum(k=1,fibonacci(n+1),eulerphi(k))-1 \\ Charles R Greathouse IV, Oct 07 2016

a(n) = 2 * A176499(n) - 3.

a(26)-a(28) from Sameen Ahmed Khan, May 02 2010

a(29)-a(30) from Antoine Mathys, Aug 06 2018

A176501 a(n) = Farey(m; I) where m = Fibonacci(n + 1) and I = [1/n, 1].

Original entry on oeis.org

1, 2, 4, 9, 19, 50, 122, 317, 837, 2213, 5758, 15236, 40028, 105079, 276627, 727409, 1910685, 5020094, 13180380, 34600740, 90814431, 238288480, 625111687, 1639676484, 4300183922, 11275936787, 29564497466, 77507123132, 203175049457, 532552499826, 1395790412496

Offset: 1

Sameen Ahmed Khan, Apr 21 2010

This sequence arises in the analytically obtained strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. This sequence provides a better strict upper bound than A176499 but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]
From Hugo Pfoertner, Jan 24 2021: (Start)
The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is wrong. It only applies to purely serial-parallel networks, but it already fails when bridges are allowed, as described in A174283. Even more so if arbitrary nonplanar networks are allowed as in A337517. See the linked illustrations of the respective quotients.
But in contrast to A176499, which at least correctly bounds A048211, the terms a(5), ..., a(9) in this sequence are smaller than the corresponding terms from A048211 (a(n) vs. A048211(n): 19/22, 50/53, 122/131, 317/337, 837/869). (End)

			n = 5, I = [1/5, 1], m = Fibonacci(5 + 1) = 8, Farey(8) = 23, Farey(8; I) = 19

Antoine Mathys, Table of n, a(n) for n = 1..40
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
Sameen Ahmed Khan, Mathematica notebook
Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).
Hugo Pfoertner, Ratio for networks with bridges, Plot2 of A174283(n)/a(n).
Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176502, A337517.

a[n_ /; n<4] := 2^(n-1); a[n_] := Module[{m = Fibonacci[n+1], v}, v = Reap[ Do[Sow[j/i], {i, n+1, m}, {j, 1, (i-1)/n}]][[2, 1]]; Total[ EulerPhi[ Range[m]]] - Length[v // Union]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* Jean-François Alcover, Aug 30 2018, after Antoine Mathys *)

farey(n) = sum(i=1, n, eulerphi(i)) + 1;
a(n) = my(m=fibonacci(n + 1), count=0); for(b=n+1, m, for(a=1, (b-1)/n, if(gcd(a,b)==1, count++))); farey(m) - 1 - count; \\ Antoine Mathys, May 07 2019

a(19)-a(27) from Antoine Mathys, Aug 10 2018

a(28)-a(31) from Antoine Mathys, May 07 2019

A337516 Number of distinct resistances that can be produced using n unit resistors in series, parallel, bridge or fork configurations.

Original entry on oeis.org

1, 2, 4, 9, 23, 57, 151, 421, 1202, 3397, 9498, 25970, 70005, 187259, 500061

Offset: 1

Rainer Rosenthal, Oct 29 2020

Each network with 2, 3 or 4 resistors is made up of series or parallel connected resistors in such a way that the resulting resistances can be computed as Ser(x1,x2) = x1 + x2 (type S) or Par(x1,x2) = 1/(1/x1+1/x2) (type P). The parameters are either 1 Ohm or themselves of type S or P. A048211 counts the different resistances which can be produced as S or P type from n unit resistors. With 5 resistors x1 .. x5 there is the bridge configuration (type B),
        A         which cannot be computed by functions Ser() and Par().
       / \        The resistance between A and D is given by
     x1   x2
     /     \      Bri(x1,x2,x3,x4,x5) =
    B- x3 - C
     \     /      x2*x1*x4+x2*x1*x5+x5*x4*x1+x5*x4*x2+x3*(x2+x5)*(x1+x4)
     x4   x5      ------------------------------------------------------ .
       \ /                  (x1+x2)*(x4+x5)+x3*(x1+x4+x2+x5)
        D
Sequence A174283 counts all resistances of types S, P and B which can be produced with n unit resistors. The next essentially new figuration comes with 7 resistors: the fork (type F), which cannot be computed by functions Ser(), Par() and Bri().
              A
             / \
           x3   x1
           /     \
          B- x5 - C
         / \     /
       x4  x7   x6
       /     \ /
      E- x2 - D
The resistance between A and E is given by
Frk(x1,x2,x3,x4,x5,x6,x7) =
x1*x3*x4*x7+x1*x3*x4*x5+x1*x3*x2*x7+x1*x3*x2*x5+x2*x4*x3*x7+x2*x4*x3*x5+
x2*x4*x1*x7+x2*x4*x1*x5+x5*x7*x1*x3+x5*x7*x1*x4+x5*x7*x2*x3+x5*x7*x2*x4+
x6*x1*x3*x7+x6*x1*x3*x2+x6*x1*x3*x4+x6*x5*x7*x3+x6*x5*x2*x3+x6*x3*x4*x5+
x6*x3*x4*x7+x6*x1*x4*x7+x6*x5*x7*x4+x6*x2*x4*x3+x6*x2*x4*x1+x6*x5*x2*x4
------------------------------------------------------------------------ .
   x3*x4*x7+x3*x4*x5+x2*x3*x7+x5*x2*x3+x1*x4*x7+x5*x1*x4+x1*x2*x7+
   x1*x2*x5+x5*x7*x3+x5*x7*x4+x5*x7*x1+x5*x7*x2+x6*x3*x7+x6*x2*x3+
   x6*x3*x4+x6*x1*x7+x6*x1*x2+x6*x1*x4+x6*x5*x7+x6*x5*x2+x6*x4*x5
This sequence A337516 counts all resistances of type S, P, B or F which can be produced with n unit resistors.

			a(1) through a(6) are identical with A174283 since a fork needs at least 7 resistors. a(7) is also equal to A174283(7) because the fork with 7 unit resistors has resistance 8/7, but this is already an element of SetA174283(7).
a(8) = 421 has six extra resistances {16/17, 40/29, 35/34, 37/29, 35/31, 37/32} which are the result of resistance 2 or 1/2 as any of the resistances x1 .. x7 except for x6.

Hugo Pfoertner, Increase in the number of representable resistance values through the fork bridge type, Plot2 of a(n)/A174283(n).
Rainer Rosenthal, Maple program SetA337516 to generate the sets counted by A337516

Cf. A048211, A174283, A337517.

# SetA337516(n) is the set of resistances counted by A337516(n) (see Maple link).
A337516 := n -> nops(SetA337516(n)):
seq(A337516(n), n=1..9);

cp's OEIS Frontend

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

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A174283 Number of distinct resistances that can be produced using n equal resistors in, series, parallel and/or bridge configurations.

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A338197 a(n) is the number of distinct resistances that can be obtained by a network of exactly n equal resistors, but not by any network with fewer than n equal resistors.

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A174286 Number of distinct resistances that can be produced using at most n equal resistors in series and/or parallel, confined to the five arms (four arms and the diagonal) of a bridge configuration. Since the bridge requires a minimum of five resistors, the first four terms are zero.

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A176502 a(n) = 2*Farey(m; I) - 1 where m = Fibonacci (n + 1) and I = [1/n, 1].

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A338600 a(n) is the common denominator of the A338197(n) rational resistance values that can be obtained from a network of exactly n one-ohm resistors, but not by a network of fewer than n one-ohm resistors.

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A176499 Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).

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