A212045 -id:A212045 - cp's OEIS frontend

A357113 T(n,m) is the numerator of the resistance between two diametrically opposite nodes of a rectangular electric network of n*m quadratic cells in which all edges are replaced by one-ohm resistors, where T(n,m) is a triangle read by rows.

1, 7, 3, 15, 121, 13, 45, 430, 2089, 47, 43, 1047, 37873, 2749, 1171, 239, 7148, 321249, 10499426, 2905619, 6385, 433, 33647, 59557, 156300899, 9176362943, 766114047605, 982871, 1157, 13971, 15887065, 1637345324, 120912032349, 25420198613182, 771357156007, 441083

Offset: 1

Hugo Pfoertner, Sep 15 2022

			The triangle of resistances begins:
   1;
   7/5,       3/2;
  15/8,     121/69,      13/7;
  45/19,   430/209,   2089/1023,     47/22;
  43/15,  1047/440,  37873/16744,  2749/1205,  1171/495
.
  O- 1 ohm -O
  |         |
  O-===-O   |
  #     #   |
  #     #   |
  O-===-O---'
.
  O-- 7/5 ohms ---O        O--- 3/2 ohms --O
  |               |        |               |
  O-===-O-===-O   |        O-===-O-===-O   |
  #     #     #   |        #     #     #   |
  #     #     #   |        #     #     #   |
  O-===-O-===-O---'        O-===-O-===-O   |
                           #     #     #   |
                           #     #     #   |
                           O-===-O-===-O---'
.
  O---- 15/8 ohms ------O  O--- 121/69 ohms -----O  O--- 13/7 ohms -------O
  |                     |  |                     |  |                     |
  O-===-O-===-O-===-O   |  O-===-O-===-O-===-O   |  O-===-O-===-O-===-O   |
  #     #     #     #   |  #     #     #     #   |  #     #     #     #   |
  #     #     #     #   |  #     #     #     #   |  #     #     #     #   |
  O-===-O-===-O-===-O---'  O-===-O-===-O-===-O   |  O-===-O-===-O-===-O   |
                           #     #     #     #   |  #     #     #     #   |
                           #     #     #     #   |  #     #     #     #   |
                           O-===-O-===-O-===-O---'  O-===-O-===-O-===-O   |
                                                    #     #     #     #   |
                                                    #     #     #     #   |
                                                    O-===-O-===-O-===-O---'

MingKun Yue, Rows n=1..24 of triangle, flattened.

A357114 are the corresponding denominators.

Cf. A093652, A212045, A212046, A357115, A357116.

ResistanceDistance[g_Graph,i_Integer,j_Integer]:=Module[{n=VertexCount[g]},ResistanceDistanceMatrix=PseudoInverse[KirchhoffMatrix[g]+ConstantArray[1/n,{n,n}]];ResistanceDistanceMatrix[[i,i]]+ResistanceDistanceMatrix[[j,j]]-ResistanceDistanceMatrix[[i,j]]-ResistanceDistanceMatrix[[j,i]]]; a[n_Integer,m_Integer]:=ResistanceDistance[GridGraph[{n,m}],1,n*m]; Numerator[Flatten[Table[a[n,m],{n,2,10},{m,2,n}]]] (* MingKun Yue, Jan 25 2025 *)

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.

Original entry on oeis.org

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

Peter J. C. Moses, Apr 30 2012

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).

			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.

F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

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

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

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]]

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.

cp's OEIS Frontend

A357113 T(n,m) is the numerator of the resistance between two diametrically opposite nodes of a rectangular electric network of n*m quadratic cells in which all edges are replaced by one-ohm resistors, where T(n,m) is a triangle read by rows.

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