A212045 - OEIS

%I #26 Apr 28 2014 09:56:12

%S 1,3,1,7,3,5,15,7,61,2,31,15,241,25,8,21,31,131,101,137,13,127,21,12,

%T 7,2381,343,151,255,127,2105,167,10781,2033,32663,32,511,255,16531,

%U 929,42061,9383,84677,2357,83,1023,511,5231,7387,74189,1771,12419

%N 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.

%C 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 R(k,n)=A212045(k,n)/A212046(k,n).  Moreover,

%C A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),

%C A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).

%D F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

%H D. J. Klein, <a href="http://dx.doi.org/10.1007/BF01164627">Resistance Distance</a>, Journal of Mathematical Chemistry 12 (1993) 81-95.

%H D. J. Klein, <a href="http://public.carnet.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf">Resistance-Distance Sum Rules</a>, Croatia Chemica Acta, 75 (2002), 633-649.

%H Nicholas Pippenger, <a href="http://arxiv.org/abs/0904.1757v1"> The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements</a>, arXiv:0904.1757v1 [math.CO], 2009.

%H 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 David Randall, <a href="http://www.radioelectronicschool.net/files/downloads/resistor_cube_problem.pdf">The Resistor Cube Problem</a>.

%H 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.

%F A212045(n)/A212046(n) is the rational number R(k, n) =

%F [(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.

%e First six rows of A212045/A212046:

%e 1

%e 3/4 .... 1

%e 7/12 ... 3/4 .... 5/6

%e 15/32 .. 7/12 ... 61/96 ... 2/3

%e 31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15

%e 21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30

%e 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 R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;

%t R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)

%t t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]

%t Flatten[Numerator[t]]    (* A212045 *)

%t Flatten[Denominator[t]]  (* A212046 *)

%t TableForm[Numerator[t]]

%t TableForm[Denominator[t]]

%Y Cf. A212046, A046878, A046879, A046825, A090634, A090633.

%K nonn,frac,tabl

%O 1,2

%A _Peter J. C. Moses_, Apr 28 2012