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
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
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 *)