cp's OEIS Frontend

A212046 Denominators 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.

%I A212046 #27 Sep 15 2022 09:20:33
%S A212046 1,4,1,12,4,6,32,12,96,3,80,32,480,48,15,64,80,320,240,320,30,448,64,
%T A212046 35,20,6720,960,420,1024,448,7168,560,35840,6720,107520,105,2304,1024,
%U A212046 64512,3584,161280,35840,322560,8960,315,5120,2304,23040,32256
%N A212046 Denominators 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.
%C A212046 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
%C A212046 R(k,n)=A212045(k,n)/A212046(k,n).  Moreover,
%C A212046 A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),
%C A212046 A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).
%D A212046 F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
%H A212046 D. J. Klein, <a href="http://dx.doi.org/10.1007/BF01164627">Resistance Distance</a>, Journal of Mathematical Chemistry 12 (1993) 81-95.
%H A212046 D. J. Klein, <a href="https://hrcak.srce.hr/file/188316">Resistance-Distance Sum Rules</a>, Croatia Chemica Acta, Vol. 75, No. 2 (2002), 633-649.
%H A212046 Nicholas Pippenger, <a href="http://arxiv.org/abs/0904.1757">The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements</a>, arXiv:0904.1757 [math.CO], 2009.
%H A212046 N. Pippenger, <a href="http://www.jstor.org/stable/10.4169/002557010X529752">The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements</a>, Mathematics Magazine, 83:5 (2010) 331-346.
%H A212046 D. Singmaster, <a href="http://dx.doi.org/10.1137/1022099">Problem 79-16, Resistances in an n-Dimensional Cube</a>, SIAM Review, 22 (1980) 504.
%H A212046 Wikipedia, <a href="http://en.wikipedia.org/wiki/Resistance_distance">Resistance distance</a>
%F A212046 A212045(n)/A212046(n) is the rational number R(k, n) =
%F A212046 [(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.
%e A212046 First six rows of A212045/A212046:
%e A212046   1
%e A212046   3/4 .... 1
%e A212046   7/12 ... 3/4 .... 5/6
%e A212046   15/32 .. 7/12 ... 61/96 ... 2/3
%e A212046   31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15
%e A212046   21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30
%e A212046 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.
%t A212046 R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;
%t A212046 R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)
%t A212046 t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]
%t A212046 Flatten[Numerator[t]]    (* A212045 *)
%t A212046 Flatten[Denominator[t]]  (* A212046 *)
%t A212046 TableForm[Numerator[t]]
%t A212046 TableForm[Denominator[t]]
%Y A212046 Cf. A212045, A046878, A046879, A046825, A090634, A090633.
%K A212046 nonn,frac,tabl
%O A212046 1,2
%A A212046 _Peter J. C. Moses_, Apr 30 2012