OFFSET
1,2
LINKS
MingKun Yue, Rows n=1..24 of triangle, flattened.
EXAMPLE
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---'
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 *)
CROSSREFS
KEYWORD
AUTHOR
Hugo Pfoertner, Sep 15 2022
STATUS
approved