A048211 -id:A048211 - cp's OEIS frontend

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

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.

A174285 Number of distinct resistances that can be produced using 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, 17, 61, 235, 815, 2563, 7585, 22277, 62065, 169489, 452621, 1191617

Offset: 1

Sameen Ahmed Khan, Mar 15 2010

			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.
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.

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, A174286, A176499, A176500, A176501, A176502.

See link section: A174285(n) = nops(SetA174285(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 04 2021
a(12) corrected by Marx Stampfli, Nov 04 2022

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);

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.

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

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).

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

cp's OEIS Frontend

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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A174285 Number of distinct resistances that can be produced using 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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A176499 Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).

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

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

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

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