A051389 -id:A051389 - 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

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

Original entry on oeis.org

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

A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

Original entry on oeis.org

1, 2, 5, 8, 8, 13, 151, 256, 83, 146, 1433, 2588, 15341, 28211, 52235, 19456, 19345, 36362, 651745, 6168632, 1463914, 2786599, 122289917, 233836352, 140001721, 268709146, 774885169, 1491969394, 41711914513, 80530073893

Offset: 0

N. J. A. Sloane

The term a(12)=15341 is divisible by 23^2. Is there another term a(n) divisible by the square of a prime p larger than n+1? - M. F. Hasler, Jul 17 2012

			1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.
R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed. Exercise 5.100.
G. Letac, Problèmes de probabilités, Presses Universitaires de France (1970), p. 14.
F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

T. D. Noe, Table of n, a(n) for n = 0..200
T. Mansour, Gamma function, beta function and combinatorial identities.
T. Sillke, More information
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Combinatorics, 14 (1993), 351-353.

Cf. A003149, A046826, A048211, A051389, A100512, A100513.

[Numerator((&+[1/Binomial(n,j): j in [0..n]])): n in [0..40]]; // G. C. Greubel, May 24 2021

Numerator/@Table[Sum[1/Binomial[n,k],{k,0,n}],{n,0,40}]  (* Harvey P. Dale, Apr 21 2011 *)

P=1;vector(30,n,numerator(P)+0*P=P/2/n*(n+1)+1) \\ M. F. Hasler, Jul 17 2012

A046825(n)=numerator(sum(k=0,n,1/binomial(n,k))) \\ M. F. Hasler, Jul 19 2012

[numerator(sum(1/binomial(n,j) for j in (0..n))) for n in (0..40)] # G. C. Greubel, May 24 2021

Let P(n) = (1/n) * Sum_{k=0..n-1} 1/binomial(n-1, k) = A046878(n)/A046879(n) = A046825(n-1)/(n*A046826(n-1)): { 0, 1, 1, 5/6, 2/3, 8/15, ...}. Then P(n) = 2^(-n) * Sum_{k=1..n} 2^k / k = 2^(-n+1) * Sum_{k odd} binomial(n, k)/k; P(0) = 0, P(n) = P(n-1)/2 + 1/n. - Torsten Sillke (Torsten.Sillke(AT)uni-bielefeld.de)
G.f. for P(n): (2*log(1-z))/(-2+z). - Wouter Meeussen
P(n) = 2^(-n) * Sum_{k=1..n} (binomial(n,k) + 1)/k.
a(n) = numerator( A003149(n)/n! ). - G. C. Greubel, May 24 2021

References entries (Comtet, Graham et al., Letac, Nedemeyer) and Links entries (Singmaster, Sury) from Torsten.Sillke(AT)uni-bielefeld.de

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

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.

cp's OEIS Frontend

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

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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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A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

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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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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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A338595 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 5 one-ohm resistors, but not from any network with fewer than 5 one-ohm resistors. Numerators are in A338580.

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A338605 Resistance values R < 1 ohm, multiplied by their common denominator 840 (= A338600(5)), that can be obtained from a network of exactly 5 one-ohm resistors, but not from any network with fewer than 5 one-ohm resistors.

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