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A212045
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.
3
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
OFFSET
1,2
COMMENTS
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
R(k,n)=A212045(k,n)/A212046(k,n). Moreover,
A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),
A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).
REFERENCES
F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
LINKS
D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 81-95.
D. J. Klein, Resistance-Distance Sum Rules, Croatia Chemica Acta, 75 (2002), 633-649.
Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757v1 [math.CO], 2009.
N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331-346.
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
FORMULA
A212045(n)/A212046(n) is the rational number R(k, n) =
[(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.
EXAMPLE
First six rows of A212045/A212046:
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.
MATHEMATICA
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]]
CROSSREFS
KEYWORD
nonn,frac,tabl
AUTHOR
Peter J. C. Moses, Apr 28 2012
STATUS
approved