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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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, 75 (2002), 633-649.

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) 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: A145369 A145370 A130322 * 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

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Last modified March 25 19:24 EDT 2017. Contains 284082 sequences.