

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 ndimensional cube.


2



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
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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 ndimensional 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) 1215 and 63.


LINKS

Table of n, a(n) for n=1..49.
D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 8195.
D. J. Klein, ResistanceDistance Sum Rules, Croatia Chemica Acta, 75 (2002), 633649.
Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757.
N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331346.
D. Singmaster, Problem 7916, Resistances in an nDimensional Cube, SIAM Review, 22 (1980) 504.
Wikipedia, Resistance distance


FORMULA

A212045(n)/A212046(n) is the rational number R(k, n) =
[(k1)*R(k2,n)n*R(k1,n)+2^(1n)]/(kn1), 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: A145370 A130322 A316232 * A232013 A246943 A106194
Adjacent sequences: A212043 A212044 A212045 * A212047 A212048 A212049


KEYWORD

nonn,frac,tabl


AUTHOR

Peter J. C. Moses, Apr 30 2012


STATUS

approved



