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

Offset: 2

Author

Hugo Pfoertner, Dec 12 2020

Keywords

• nonn
• hard
• more

Examples

			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]
		

Links

• Fedor Karpelevitch, Program output: lower.txt
• Fedor Karpelevitch, Program, run with `./gradlew cir2 --args=best`

Crossrefs

Cf. A180414, A337517, A339808.

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

Extensions

a(18) from Hugo Pfoertner, Apr 09 2021

a(19) from Fedor Karpelevitch, Aug 17 2025