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A048211
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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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29
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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a(2) = 2 since given two 1-ohm resistors, a series circuit yields 2 ohms, while a parallel circuit yields 1/2 ohms.
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MAPLE
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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)):
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MATHEMATICA
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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 *)
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PROG
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(PARI) \\ 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], ", ") );
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CROSSREFS
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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
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KEYWORD
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nonn,nice,more,hard
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AUTHOR
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EXTENSIONS
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Definition edited (to specify that the sequence considers only series and parallel combinations) by Jon E. Schoenfield, Sep 02 2013
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STATUS
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approved
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