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

Author

Hugo Pfoertner, Sep 15 2022

Keywords

• nonn
• frac
• tabl

Examples

			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---'
		

Links

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

Crossrefs

A357114 are the corresponding denominators.

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

Programs

Mathematica

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 *)