A176497 -id:A176497 - cp's OEIS frontend

A048211 Number of distinct resistances that can be produced from a circuit of n equal resistors using only series and parallel combinations.

1, 2, 4, 9, 22, 53, 131, 337, 869, 2213, 5691, 14517, 37017, 93731, 237465, 601093, 1519815, 3842575, 9720769, 24599577, 62283535, 157807915, 400094029, 1014905643, 2576046289, 6541989261, 16621908599, 42251728111, 107445714789, 273335703079

Offset: 1

Tony Bartoletti

Found by exhaustive search. Program produces all values that are combinations of two binary operators a() and b() (here "sum" and "reciprocal sum of reciprocals") over n occurrences of 1. E.g., given 4 occurrences of 1, the code forms all allowable postfix forms, such as 1 1 1 1 a a a and 1 1 b 1 1 a b, etc. Each resulting form is then evaluated according to the definitions for a and b.
Each resistance that can be constructed from n 1-ohm resistors in a circuit can be written as the ratio of two positive integers, neither of which exceeds the (n+1)st Fibonacci number. E.g., for n=4, the 9 resistances that can be constructed can be written as 1/4, 2/5, 3/5, 3/4, 1/1, 4/3, 5/3, 5/2, 4/1 using no numerator or denominator larger than Fib(n+1) = Fib(5) = 5. If a resistance x can be constructed from n 1-ohm resistors, then a resistance 1/x can also be constructed from n 1-ohm resistors. - Jon E. Schoenfield, Aug 06 2006
The fractions in the comment above are a superset of the fractions occurring here, corresponding to the upper bound A176500. - Joerg Arndt, Mar 07 2015
The terms of this sequence consider only series and parallel combinations; A174283 considers bridge combinations as well. - Jon E. Schoenfield, Sep 02 2013

			a(2) = 2 since given two 1-ohm resistors, a series circuit yields 2 ohms, while a parallel circuit yields 1/2 ohms.

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). [From _Sameen Ahmed Khan_, Apr 27 2010]
Sameen Ahmed Khan, Mathematica program
Sameen Ahmed Khan, Mathematica notebook for A048211 and A000084
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346 [physics.gen-ph], 2010.
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
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.
Marx Stampfli, Bridged graphs, circuits and Fibonacci numbers, Applied Mathematics and Computation, Volume 302, 1 June 2017, Pages 68-79.

Let T(x, n) = 1 if x can be constructed with n 1-ohm resistors in a circuit, 0 otherwise. Then A048211 is t(n) = sum(T(x, n)) for all x (x is necessarily rational). Let H(x, n) = 1 if T(x, n) = 1 and T(x, k) = 0 for all k < n, 0 otherwise. Then A051389 is h(n) = sum(H(x, n)) for all x (x is necessarily rational).
Cf. A153588, A174283, A174284, A174285 and A174286, A176497, A176498, A176499, A176500, A176501, A176502. - Sameen Ahmed Khan, Apr 27 2010
Cf. A180414.

r:= proc(n) option remember; `if`(n=1, {1}, {seq(seq(seq(
      [f+g, 1/(1/f+1/g)][], g in r(n-i)), f in r(i)), i=1..n/2)})
    end:
a:= n-> nops(r(n)):
seq(a(n), n=1..15);  # Alois P. Heinz, Apr 02 2015

r[n_] := r[n] = If[n == 1, {1}, Union @ Flatten @ {Table[ Table[ Table[ {f+g, 1/(1/f+1/g)}, {g, r[n-i]}], {f, r[i]}], {i, 1, n/2}]}]; a[n_] := Length[r[n]]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

\\ not efficient; just to show the method
N=10;
L=vector(N);  L[1]=[1];
{ for (n=2, N,
    my( T = Set( [] ) );
    for (k=1, n\2,
        for (j=1, #L[k],
            my( r1 = L[k][j] );
            for (i=1, #L[n-k],
                my( r2 = L[n-k][i] );
                T = setunion(T,  Set([r1+r2, r1*r2/(r1+r2) ]) );
            );
        );
    );
    T = vecsort(Vec(T), , 8);
    L[n] = T;
); }
for(n=1, N, print1(#L[n], ", ") );
\\ Joerg Arndt, Mar 07 2015

From Bill McEachen, Jun 08 2024: (Start)
(2.414^n)/4 < a(n) < (1-1/n)*(0.318)*(2.618^n) (Khan, n>3).
Conjecture: a(n) ~ K * a(n-1), K approx 2.54. (End)

More terms from John W. Layman, Apr 06 2002
a(16)-a(21) from Jon E. Schoenfield, Aug 06 2006
a(22) from Jon E. Schoenfield, Aug 28 2006
a(23) from Jon E. Schoenfield, Apr 18 2010
Definition edited (to specify that the sequence considers only series and parallel combinations) by Jon E. Schoenfield, Sep 02 2013
a(24)-a(25) from Antoine Mathys, Apr 02 2015
a(26)-a(27) from Johannes P. Reichart, Nov 24 2018
a(28)-a(30) from Antoine Mathys, Dec 08 2024

A153588 Number of resistance values that can be constructed using up to n equal resistances by arranging them in an arbitrary series-parallel arrangement.

Original entry on oeis.org

1, 3, 7, 15, 35, 77, 179, 429, 1039, 2525, 6235, 15463, 38513, 96231, 241519, 607339, 1529533, 3857447, 9743247, 24634043, 62335495, 157885967, 400211085, 1015080877, 2576308943, 6542380707, 16622493939, 42252603207, 107447022475, 273337662943

Offset: 1

Altrego Janeway (altrego99(AT)gmail.com), Dec 29 2008

			For n=2 there are 3 solutions, 1 ohm, (1+1) ohms and 1/(1/1+1/1)=1/2 ohm. So a(2)=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).
Sameen Ahmed Khan, Mathematica notebook for A153588 and A058351
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346 [physics.gen-ph], 2010.
Sameen Ahmed Khan, How Many Equivalent Resistances?, RESONANCE, May 2012.
Sameen Ahmed Khan, Beginning to count the number of equivalent resistances, Indian Journal of Science and Technology, 2016, Vol 9(44).
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.
Project Euler, Problem 155: Counting Capacitor Circuits.

Cf. A048211. This sequence is the total number of resistance values formed using up to n resistances, A048211 is the total number of resistance values formed using exactly n resistances.

Cf. A048211, A153588, A174283, A174284, A174285 and A174286, A176497, A176498, A176499, A176500, A176501, A176502. [Sameen Ahmed Khan, Apr 27 2010]

a(17)-a(25) from Antoine Mathys, Apr 02 2015
Definition clarified by Antoine Mathys, Apr 03 2015
a(26)-a(30) from Antoine Mathys, Dec 08 2024

A176498 Number of elements less than 1/2 in the Cross Set which is the subset of the set of distinct resistances that can be produced using n unit resistors in series and/or parallel.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 9, 24, 58, 124, 312, 759, 1768, 4421, 10811, 27191, 68591, 174627, 441633, 1124795, 2866004, 7297500, 18585359, 47337643, 120562041, 307063757

Offset: 1

Sameen Ahmed Khan, Apr 21 2010

This sequence arises in the decomposition of the sets A(n + 1) of equivalent resistances, when n equal resistors are combined in series/parallel, into series parallel and cross sets respectively. All the elements of the parallel set are strictly less than 1 and all those of the series set are strictly greater than 1. The cross set is expected to be dense around 1 with very few elements below 1/2. Hence it is relevant to count the elements below 1/2.

			The order of the cross set is given by A176497: 0, 0, 0, 1, 4, 9, 25, 75, 195, 475, 1265, 3135, ... The sets corresponding n = 4 to n = 8 do not have a single element below 1/2. For n = 9 onwards we have a few elements which are less than 1/2; they are 1, 6, 9, 24, ....

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).
S. A. Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - _N. J. A. Sloane_, Oct 23 2012

Cf. A048211, A176497.

a(16)-a(25) from Antoine Mathys, Mar 20 2017

a(26)-a(30) from Antoine Mathys, Dec 08 2024

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A048211 Number of distinct resistances that can be produced from a circuit of n equal resistors using only series and parallel combinations.

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A153588 Number of resistance values that can be constructed using up to n equal resistances by arranging them in an arbitrary series-parallel arrangement.

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A176498 Number of elements less than 1/2 in the Cross Set which is the subset of the set of distinct resistances that can be produced using n unit resistors in series and/or parallel.

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