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A357114 T(n,m) is the denominator of the resistance between two diametrically opposite nodes of a rectangular electric network of n*m quadratic meshes in which all edges are replaced by one-ohm resistors, where T(n,m) is a triangle read by rows.

%I A357114 #12 Jan 29 2025 08:19:23
%S A357114 1,5,2,8,69,7,19,209,1023,22,15,440,16744,1205,495,71,2639,128617,
%T A357114 4282081,1169441,2494,112,11067,21728,59292739,3498175408,
%U A357114 287916805961,360161,265,4142,5317209,579080689,43600867640,9153575734849,273893674761,153254
%N A357114 T(n,m) is the denominator of the resistance between two diametrically opposite nodes of a rectangular electric network of n*m quadratic meshes in which all edges are replaced by one-ohm resistors, where T(n,m) is a triangle read by rows.
%H A357114 MingKun Yue, <a href="/A357114/b357114.txt">Rows n=1..24 of triangle, flattened</a>
%e A357114 The triangle begins:
%e A357114    1;
%e A357114    7/5,       3/2;
%e A357114   15/8,     121/69,      13/7;
%e A357114   45/19,   430/209,   2089/1023,     47/22;
%e A357114   43/15,  1047/440,  37873/16744,  2749/1205,  1171/495
%t A357114 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]; Denominator[Flatten[Table[a[n,m],{n,2,10},{m,2,n}]]] (* _MingKun Yue_, Jan 25 2025 *)
%Y A357114 A357113 are the corresponding numerators.
%K A357114 nonn,frac,tabl
%O A357114 1,2
%A A357114 _Hugo Pfoertner_, Sep 15 2022