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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.

%I A357113 #18 Feb 07 2025 21:22:39
%S A357113 1,7,3,15,121,13,45,430,2089,47,43,1047,37873,2749,1171,239,7148,
%T A357113 321249,10499426,2905619,6385,433,33647,59557,156300899,9176362943,
%U A357113 766114047605,982871,1157,13971,15887065,1637345324,120912032349,25420198613182,771357156007,441083
%N 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.
%H A357113 MingKun Yue, <a href="/A357113/b357113.txt">Rows n=1..24 of triangle, flattened</a>.
%e A357113 The triangle of resistances begins:
%e A357113    1;
%e A357113    7/5,       3/2;
%e A357113   15/8,     121/69,      13/7;
%e A357113   45/19,   430/209,   2089/1023,     47/22;
%e A357113   43/15,  1047/440,  37873/16744,  2749/1205,  1171/495
%e A357113 .
%e A357113   O- 1 ohm -O
%e A357113   |         |
%e A357113   O-===-O   |
%e A357113   #     #   |
%e A357113   #     #   |
%e A357113   O-===-O---'
%e A357113 .
%e A357113   O-- 7/5 ohms ---O        O--- 3/2 ohms --O
%e A357113   |               |        |               |
%e A357113   O-===-O-===-O   |        O-===-O-===-O   |
%e A357113   #     #     #   |        #     #     #   |
%e A357113   #     #     #   |        #     #     #   |
%e A357113   O-===-O-===-O---'        O-===-O-===-O   |
%e A357113                            #     #     #   |
%e A357113                            #     #     #   |
%e A357113                            O-===-O-===-O---'
%e A357113 .
%e A357113   O---- 15/8 ohms ------O  O--- 121/69 ohms -----O  O--- 13/7 ohms -------O
%e A357113   |                     |  |                     |  |                     |
%e A357113   O-===-O-===-O-===-O   |  O-===-O-===-O-===-O   |  O-===-O-===-O-===-O   |
%e A357113   #     #     #     #   |  #     #     #     #   |  #     #     #     #   |
%e A357113   #     #     #     #   |  #     #     #     #   |  #     #     #     #   |
%e A357113   O-===-O-===-O-===-O---'  O-===-O-===-O-===-O   |  O-===-O-===-O-===-O   |
%e A357113                            #     #     #     #   |  #     #     #     #   |
%e A357113                            #     #     #     #   |  #     #     #     #   |
%e A357113                            O-===-O-===-O-===-O---'  O-===-O-===-O-===-O   |
%e A357113                                                     #     #     #     #   |
%e A357113                                                     #     #     #     #   |
%e A357113                                                     O-===-O-===-O-===-O---'
%t A357113 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 *)
%Y A357113 A357114 are the corresponding denominators.
%Y A357113 Cf. A093652, A212045, A212046, A357115, A357116.
%K A357113 nonn,frac,tabl
%O A357113 1,2
%A A357113 _Hugo Pfoertner_, Sep 15 2022