

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.


3



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, Vol. 75, No. 2 (2002), 633649.
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) 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 A353791
Adjacent sequences: A212043 A212044 A212045 * A212047 A212048 A212049


KEYWORD

nonn,frac,tabl


AUTHOR

Peter J. C. Moses, Apr 30 2012


STATUS

approved



