A176497
a(n) is the cardinality of the "Cross Set" which is the subset of distinct resistances that can be produced by a circuit of n unit resistors using only series or parallel combinations which cannot be decomposed as a single unit resistor in either series or parallel with a circuit of n-1 unit resistors.
Original entry on oeis.org
0, 0, 0, 1, 4, 9, 25, 75, 195, 475, 1265, 3135, 7983, 19697, 50003, 126163, 317629, 802945, 2035619, 5158039, 13084381, 33240845, 84478199, 214717585, 546235003, 1389896683, 3537930077, 9007910913, 22942258567, 58444273501
Offset: 1
A(1) has no cross set and the first term is defined to be zero; the cross sets for n = 2 and n = 3 are empty hence the second and third term are each zero. Noting that A(3) = 4 and A(4) = 9, the fourth term is 1. The fifth term is 4.
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
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
A338401
a(n) is the numerator of the resistance R(n) = a(n)/A338402(n) of a triangular network of 3*n*(n+1)/2 one Ohm resistors in a hexagonal lattice arrangement.
Original entry on oeis.org
2, 10, 10, 206, 3326, 43118, 150806, 11591578, 436494606, 1008712015454, 382034633808890, 13187511533010430, 2111825680430510462, 171204772756285452656378, 89579048665281690355286, 1013412795315891086553473628734, 20023655015717377508089133638478, 24678955315461926144059519221489609194
Offset: 1
R(1) = a(1)/A338402(1) = 2/3,
R(2) = a(2)/A338402(2) = 10/9,
R(4) = a(4)/A338402(4) = 206/123.
a(3) = 10: The following network of A045943(3) = 18 one Ohm resistors has a resistance of R(3) = 10/7 Ohm, i.e., the current I driven by the voltage of 1 Volt is 7/10 = A338402(3)/a(3) Ampere.
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O
__/ \_
/ / \ \
/1/ \1\
/_/ \_\
/ _____ \
O---|__1__|---O
__/ \_ __/ \_
/ / \ \ / / \ \
/1/ \1\ /1/ \1\
/_/ \_\ /_/ \_\
/ _____ \ / _____ \
O---|__1__|---O---|__1__|---O
__/ \__ __/ \__ __/ \_
/ / \ \ / / \ \ / / \ \
/1/ \1\ /1/ \1\ /1/ \1\
/_/ \_\ /_/ \_\ /_/ \_\
/ _____ \ / _____ \ / _____ \
O---|__1__|---O---|__1__|---O---|__1__|---O
| |
| V = 1 Volt |
| | |
-------------------| |-- I=1/R Ampere ---
|
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With a numbering of the resistors as shown in the following diagram,
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O
/ \
15 18
/ \
O--14---O
/ \ / \
7 9 13 17
/ \ / \
O-- 6---O--12---O
/ \ / \ / \
2 3 5 8 11 16
/ \ / \ / \
O---1---O---4---O--10---O
|______1 Volt__I=I19____|
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the currents in Amperes through the 18 resistors, and the current I=I19 through the voltage source of 1 Volt, are [11/30, 1/3, 1/30, 4/15, 2/15, 1/6, 2/15, 2/15, 1/30, 11/30, 1/30, 1/6, 1/30, 1/15, 1/30, 1/3, 2/15, 1/30, 7/10].
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a33840_1_2(n)={my(md=3*n*(n+1)/2+1,
T1=matrix(n,n),T2=matrix(n,n),T3=matrix(n,n),
M=matrix(md,md,i,j,0),U=vector(md),
valid(i,j)=i>0&&i<=n&&j>0&&j<=n&&i>=j,k=0,neq=1);
\\ List of edges
for(i=1,n,for(j=1,i,T1[i,j]=k++;T2[i,j]=k++;T3[i,j]=k++));
\\ In- and outflow of current at all nodes
\\ lower left triangle with inflow of current from source of voltage
M[1,1]=-1;M[1,2]=-1;M[1,md]=1;
\\ loops over lower left corners of triangles
for(i=2,n+1,for(j=1,i,
\\ exclude node at top of triangle
if(j
A340920
a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.
Original entry on oeis.org
1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3807, 11549, 34843, 104459, 311317, 928719, 2776247, 8320757, 24967341, 74985337
Offset: 0
a(10) = 3807, whereas A337517(10) = 3823. The difference of 16 resistances results from the 15 terms of A338601/A338602 and the resistance 34/27 not representable by a planar network of 10 resistors, whereas it (but not 27/34) can be represented by a nonplanar network of 10 resistors.
A338781
Half the maximum number of distinct resistances that can be produced from a circuit of n resistors of two different kinds using only series and parallel combinations.
Original entry on oeis.org
1, 3, 10, 38, 161, 718, 3385, 16548, 83183, 427490, 2237196, 11865560, 63677761
Offset: 1
In the following let x and y be the values of the two resistors.
With 1 component the resistances are {x, y}, so a(1) = 2/2 = 1.
With 2 components the resistances are {2*x, x/2, 2*y, y/2, x + y, x*y/(x + y)}, so a(2) = 6/2 = 3.
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ParSer(u,v)={concat(concat(vector(#u, i, vector(#v, j, u[i]+v[j]))), concat(vector(#u, i, vector(#v, j, 1/(1/u[i]+1/v[j])))))}
S(n)={my(v=vector(n)); v[1]=[1,'x]; for(n=2, #v, v[n]=Set(concat(vector(n\2, k, ParSer(v[k],v[n-k]))))); v}
a(n)={#(S(n)[n])/2}
A232005
Number of distinct resistances that can be produced from a circuit of resistors with resistances 1, 2, ..., n using only series and parallel combinations.
Original entry on oeis.org
1, 2, 8, 48, 386, 3781, 49475, 762869, 13554897, 266817541
Offset: 1
a(2) = 2 since given a 1-ohm and a 2-ohm resistor, a series circuit yields 3 ohms, while a parallel circuit yields 2/3 ohms, which thus yields two distinct resistances.
A292126
Number of two-terminal exclusive-bridged graphs with n edges.
Original entry on oeis.org
0, 0, 0, 0, 1, 4, 21, 86, 349, 1328, 4925, 17786
Offset: 1
A341536
Number of distinct resistances that can be produced using at most n unit resistors in series, parallel, bridge or fork configurations.
Original entry on oeis.org
1, 2, 4, 8, 16, 36, 80, 194, 500, 1342, 3623, 9835, 26412, 70505, 187805, 500627
Offset: 0
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