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A212045
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Numerators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.
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3
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1, 3, 1, 7, 3, 5, 15, 7, 61, 2, 31, 15, 241, 25, 8, 21, 31, 131, 101, 137, 13, 127, 21, 12, 7, 2381, 343, 151, 255, 127, 2105, 167, 10781, 2033, 32663, 32, 511, 255, 16531, 929, 42061, 9383, 84677, 2357, 83, 1023, 511, 5231, 7387, 74189, 1771, 12419
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OFFSET
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1,2
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COMMENTS
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The term "resistance distance" for electric circuits was in use years before it was proved to be a metric (on edges of graphs). The historical meaning has been described thus: "one imagines unit resistors on each edge of a graph G and takes the resistance distance between vertices i and j of G to be the effective resistance between vertices i and j..." (from Klein, 2002; see the References). Let R(k,n) denote the resistance distance for k resistors in an n-dimensional cube (for details, see Example and References). Then
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REFERENCES
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F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
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LINKS
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FORMULA
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[(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.
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EXAMPLE
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1
3/4 .... 1
7/12 ... 3/4 .... 5/6
15/32 .. 7/12 ... 61/96 ... 2/3
31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15
21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30
The resistance distances for n=3 (the ordinary cube) are 7/12, 3/4, and 5/6, so that row 3 of the triangle of numerators is (7, 3, 5). For the corresponding electric circuit, suppose X is a vertex of the cube. The resistance across any one of the 3 edges from X is 7/12 ohm; the resistance across any two adjoined edges (i.e., a diagonal of a face of the cubes) is 3/4 ohm; the resistance across and three adjoined edges (a diagonal of the cube) is 5/6 ohm.
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MATHEMATICA
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R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;
R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)
t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]
Flatten[Numerator[t]] (* A212045 *)
Flatten[Denominator[t]] (* A212046 *)
TableForm[Numerator[t]]
TableForm[Denominator[t]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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