cp's OEIS Frontend

A357115 T(n,m) is the numerator of the resistance between two nodes located at the end of a side of length n 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 square array read by descending antidiagonals.

%I A357115 #10 Sep 16 2022 15:38:45
%S A357115 3,11,4,41,5,13,153,26,267,26,571,68,181,192,149,2131,89,10609,1506,
%T A357115 1171,138,7953,466,25059,251,155927,246,375,29681,1220,3869723,13852,
%U A357115 759435,77948,75255,668,110771,1597,1334085,781778,109897,1949020,982871,24995,3523
%N A357115 T(n,m) is the numerator of the resistance between two nodes located at the end of a side of length n 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 square array read by descending antidiagonals.
%F A357115 Limit_{n->oo} T(1,n)/A357116(1,n) = sqrt(3) - 1.
%e A357115 The array of resistances starts:
%e A357115     3/4,      11/15,        41/56,         153/209, ... A001835(n-1)/A001353(n-1)
%e A357115     4/3,       5/4,         26/21,          68/55,   ...
%e A357115    13/7,     267/161,      181/112,      10609/6603,  ...
%e A357115    26/11,    192/95,      1506/781,        251/132,   ...
%e A357115   149/52,   1171/495,   155927/70616,   759435/352583,  ...
%e A357115   138/41,    246/91,     77948/31529,  1949020/817991,  ...
%e A357115 .
%e A357115 T(1,3)/A357116(1,3) = 41/56:
%e A357115 .      _____       _____       _____
%e A357115    O--|__1__|--O--|__1__|--O--|__1__|--O-----O
%e A357115    |           |           |           |     |
%e A357115   | |         | |         | |         | |    |
%e A357115   |1|         |1|         |1|         |1|  41/56 ohms
%e A357115   |_|         |_|         |_|         |_|    |
%e A357115    |   _____   |   _____   |   _____   |     |
%e A357115    O--|__1__|--O--|__1__|--O--|__1__|--O-----O
%e A357115 .
%e A357115 T(3,1)/A357116(3,1) = 13/7    T(2,2)/A357116(2,2) = 5/4
%e A357115 .      _____                    _____       _____
%e A357115    O--|__1__|--O-----O      O--|__1__|--O--|__1__|--O-----O
%e A357115    |           |     |      |           |           |     |
%e A357115   | |         | |    |     | |         | |         | |    |
%e A357115   |1|         |1|    |     |1|         |1|         |1|    |
%e A357115   |_|         |_|    |     |_|         |_|         |_|    |
%e A357115    |   _____   |     |      |   _____   |   _____   |     |
%e A357115    O--|__1__|--O     |      O--|__1__|--O--|__1__|--O    5/4 ohms
%e A357115    |           |     |      |           |           |     |
%e A357115   | |         | |  13/7    | |         | |         | |    |
%e A357115   |1|         |1|  ohms    |1|         |1|         |1|    |
%e A357115   |_|         |_|    |     |_|         |_|         |_|    |
%e A357115    |   _____   |     |      |   _____   |   _____   |     |
%e A357115    O--|__1__|--O     |      O--|__1__|--O--|__1__|--O-----O
%e A357115    |           |     |
%e A357115   | |         | |    |
%e A357115   |1|         |1|    |
%e A357115   |_|         |_|    |
%e A357115    |   _____   |     |
%e A357115    O--|__1__|--O-----O
%Y A357115 A357116 are the corresponding denominators.
%Y A357115 Cf. A357113, A357114.
%K A357115 nonn,frac,tabl
%O A357115 1,1
%A A357115 _Hugo Pfoertner_, Sep 15 2022