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.

3, 11, 4, 41, 5, 13, 153, 26, 267, 26, 571, 68, 181, 192, 149, 2131, 89, 10609, 1506, 1171, 138, 7953, 466, 25059, 251, 155927, 246, 375, 29681, 1220, 3869723, 13852, 759435, 77948, 75255, 668, 110771, 1597, 1334085, 781778, 109897, 1949020, 982871, 24995, 3523

Offset

1,1

Links

Table of n, a(n) for n=1..45.

Formula

Limit_{n->oo} T(1,n)/A357116(1,n) = sqrt(3) - 1.

Example

The array of resistances starts:
    3/4,      11/15,        41/56,         153/209, ... A001835(n-1)/A001353(n-1)
    4/3,       5/4,         26/21,          68/55,   ...
   13/7,     267/161,      181/112,      10609/6603,  ...
   26/11,    192/95,      1506/781,        251/132,   ...
  149/52,   1171/495,   155927/70616,   759435/352583,  ...
  138/41,    246/91,     77948/31529,  1949020/817991,  ...
.
T(1,3)/A357116(1,3) = 41/56:
.      _____       _____       _____
   O--|__1__|--O--|__1__|--O--|__1__|--O-----O
   |           |           |           |     |
  | |         | |         | |         | |    |
  |1|         |1|         |1|         |1|  41/56 ohms
  |_|         |_|         |_|         |_|    |
   |   _____   |   _____   |   _____   |     |
   O--|__1__|--O--|__1__|--O--|__1__|--O-----O
.
T(3,1)/A357116(3,1) = 13/7    T(2,2)/A357116(2,2) = 5/4
.      _____                    _____       _____
   O--|__1__|--O-----O      O--|__1__|--O--|__1__|--O-----O
   |           |     |      |           |           |     |
  | |         | |    |     | |         | |         | |    |
  |1|         |1|    |     |1|         |1|         |1|    |
  |_|         |_|    |     |_|         |_|         |_|    |
   |   _____   |     |      |   _____   |   _____   |     |
   O--|__1__|--O     |      O--|__1__|--O--|__1__|--O    5/4 ohms
   |           |     |      |           |           |     |
  | |         | |  13/7    | |         | |         | |    |
  |1|         |1|  ohms    |1|         |1|         |1|    |
  |_|         |_|    |     |_|         |_|         |_|    |
   |   _____   |     |      |   _____   |   _____   |     |
   O--|__1__|--O     |      O--|__1__|--O--|__1__|--O-----O
   |           |     |
  | |         | |    |
  |1|         |1|    |
  |_|         |_|    |
   |   _____   |     |
   O--|__1__|--O-----O

Crossrefs

A357116 are the corresponding denominators.

Cf. A357113, A357114.

Sequence in context: A226625 A210610 A188950 * A308968 A084466 A084462

Adjacent sequences: A357112 A357113 A357114 * A357116 A357117 A357118

Keyword

nonn,frac,tabl

Author

Hugo Pfoertner, Sep 15 2022

Status

approved