A212045 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.

1, 3, 1, 7, 3, 5, 15, 7, 61, 2, 31, 15, 241, 25, 8, 21, 31, 131, 101, 137, 13, 127, 21, 12, 7, 2381, 343, 151, 255, 127, 2105, 167, 10781, 2033, 32663, 32, 511, 255, 16531, 929, 42061, 9383, 84677, 2357, 83, 1023, 511, 5231, 7387, 74189, 1771, 12419

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..52.

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:0904.1757v1 [math.CO], 2009.

N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331-346.

David Randall, The Resistor Cube Problem.

D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.

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. A212046, A046878, A046879, A046825, A090634, A090633.

Sequence in context: A370689 A247675 A053092 * A292849 A115873 A329369

Adjacent sequences: A212042 A212043 A212044 * A212046 A212047 A212048

Keyword

nonn,frac,tabl

Author

Peter J. C. Moses, Apr 28 2012

Status

approved