A339548 1 - 1/a(n) is the largest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

2, 3, 4, 7, 11, 19, 35, 56, 105, 177, 321, 610, 1001, 1893, 3186, 5714, 10073, 18506, 32715

Offset

2,1

Links

Table of n, a(n) for n=2..20.

Fedor Karpelevitch, Program output: lower.txt

Fedor Karpelevitch, Program

Example

The resistor networks from which the target resistance R = 1 - 1/a(n) can be obtained correspond to simple or multigraphs whose edges are one-ohm resistors. Parallel resistors on one edge are indicated by an exponent > 1 after the affected vertex pair. The resistance R occurs between vertex number 1 and the vertex with maximum number in the graph. In some cases there are other possible representations in addition to the representation given.
.
resistors      vertices
   |     R        |  edges
   2     1/2      2 [1,2]^2
   3     2/3      3 [1,2],[1,3],[2,3]
   4     3/4      4 [1,2],[1,4],[2,3],[3,4]
   5     6/7      4 [1,2]^2,[1,3],[2,4],[3,4]
   6    10/11     5 [1,2],[1,3],[1,4],[2,3],[3,5],[4,5]
   7    18/19     5 [1,2],[1,3]^2,[2,4],[3,4],[3,5],[4,5]
   8    34/35     6 [1,2],[1,3],[1,4],[2,5],[3,4],[3,5],[4,6],[5,6]
   9    55/56     6 [1,2]^2,[1,3],[2,4],[3,5],[3,6],[4,5],[4,6],[5,6]
  10   104/105    7 [1,4],[1,5],[2,4],[2,6],[2,7],[3,5],[3,6],[3,7],[4,6],[5,7]
  11   176/177    7 [1,4],[1,6],[2,4],[2,5],[2,7],[3,5],[3,6],[3,7],[4,6],[4,7],
                    [5,7]
  12   320/321    7 [1,4],[1,6],[2,4],[2,5],[2,6],[2,7],[3,4],[3,5],[3,6],[4,6],
                    [4,7],[5,7]
  13   609/610    8 [1,4],[1,5],[1,7],[2,5],[2,6],[2,7],[3,4],[3,6],[3,7],[4,5],
                    [4,6],[6,8],[7,8]
  14  1000/1001   8 [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],[3,7],
                    [4,5],[4,6],[4,8],[6,8]
  15  1892/1893   9 [1,4],[1,5],[2,5],[2,6],[2,7],[2,9],[3,6],[3,7],[3,8],[3,9],
                    [4,7],[4,8],[4,9],[5,8],[6,8]
  16  3185/3186   9 [1,2],[1,3],[2,6],[2,7],[2,9],[3,6],[3,7],[3,8],[4,5],[4,7],
                    [4,8],[5,6],[5,8],[5,9],[6,7],[8,9]
  17  5713/5714  10 [1,2],[1,3],[2,4],[2,5],[2,7],[3,4],[3,6],[3,10],[4,8],[5,6],
                    [5,7],[5,9],[6,8],[7,8],[7,9],[8,10],[9,10]
  18 10072/10073 10 [1,2],[1,3],[2,4],[2,5],[2,6],[3,4],[3,5],[3,10],[4,8],[5,7],
                    [5,9],[6,7],[6,8],[6,9],[7,8],[7,9],[8,10],[9,10]
  19 18505/18506 11 [1,2],[1,3],[2,5],[2,6],[2,7],[3,4],[3,5],[3,11],[4,6],[4,7],
                    [5,8],[5,10],[6,8],[6,9],[7,9],[7,10],[8,9],[9,11],[10,11]
  20 32714/32715 12 [1,2],[1,3],[1,4],[2,5],[2,6],[3,7],[3,8],[3,9],[4,5],[4,7],
                    [5,8],[5,10],[6,7],[6,9],[7,11],[8,12],[9,10],[10,11],[10,12],
                    [11,12]

Crossrefs

Cf. A180414, A337517, A339808.

Cf. A279317, showing that maximum solutions using the square packing analogy can only be obtained for n <= 11 resistors.

Sequence in context: A188624 A327010 A341823 * A256994 A107481 A058521

Adjacent sequences: A339545 A339546 A339547 * A339549 A339550 A339551

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Dec 12 2020

Extensions

a(18) from Hugo Pfoertner, Apr 09 2021

a(19) from Fedor Karpelevitch, Aug 17 2025

a(20) from Fedor Karpelevitch, Sep 18 2025

Status

approved