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.

1, 4, 1, 12, 4, 6, 32, 12, 96, 3, 80, 32, 480, 48, 15, 64, 80, 320, 240, 320, 30, 448, 64, 35, 20, 6720, 960, 420, 1024, 448, 7168, 560, 35840, 6720, 107520, 105, 2304, 1024, 64512, 3584, 161280, 35840, 322560, 8960, 315, 5120, 2304, 23040, 32256

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

Table of n, a(n) for n=1..49.

D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 81-95.

D. J. Klein, Resistance-Distance Sum Rules, Croatia Chemica Acta, Vol. 75, No. 2 (2002), 633-649.

Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757 [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.

Wikipedia, Resistance distance

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

Cf. A212045, A046878, A046879, A046825, A090634, A090633.

Sequence in context: A130322 A316232 A366885 * A232013 A246943 A353791

Adjacent sequences: A212043 A212044 A212045 * A212047 A212048 A212049

Keyword

nonn,frac,tabl

Author

Peter J. C. Moses, Apr 30 2012

Status

approved