A339808 1 + 1/a(n) is the smallest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

1, 2, 3, 6, 10, 18, 34, 55, 104, 176, 320, 592, 1071, 1855, 3311, 5943, 10231

Offset

2,2

Links

Table of n, a(n) for n=2..18.

Example

a(2) = 1: 2 resistors in series produce a resistance of 2 = 1 + 1/a(1) ohm.
a(3) = 2: 3 resistors can produce {1/3, 2/3, 3/2, 3} ohms. The smallest resistance > 1 is 3/2 = 1 + 1/a(2) ohms.
a(4) = 3: 4 resistors can produce the A337517(4) = 9 distinct resistances {1/4, 2/5, 3/5, 3/4, 1, 4/3, 5/3, 5/2, 4} of which 4/3 = 1 + 1/a(4) is the smallest resistance > 1 ohm.
a(n) first differs from A339548(n) - 1 for n = 13. The resistance values of the A337517(13) = 110953 distinct resistances that can be obtained from a network of exactly 13 one-ohm resistances closest to 1 ohm are { ..., 551/552, 576/577, 596/597, 609/610, 1, 593/592, 580/579, 552/551, ...}. The largest resistance < 1 of a network of 13 one-ohm resistors is 609/610 = 1 - 1/A339548(13) ohms, whereas the smallest resistance > 1 is 593/a(13) = 593/592 ohms.
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    2/1     2 [1,2],[2,3]
   3    3/2     3 [1,2]^2,[2,3]
   4    4/3     4 [1,2]^3,[2,3]
   5    7/6     4 [1,2]^2,[2,3],[2,4],[3,4]
   6   11/10    5 [1,2]^2,[2,3]^2,[2,4],[3,4]
   7   19/18    5 [1,2]^2,[1,3],[2,3],[2,4],[3,5],[4,5]
   8   35/34    6 [1,2]^2,[1,3],[2,3],[2,4],[3,4],[3,5],[4,5]
   9   56/55    6 [1,2],[1,3],[1,4],[2,4],[3,4],[3,5],[4,5],[4,6],[5,6]
  10  105/104   7 [1,3],[1,4],[1,5],[2,4],[2,5],[2,6],[3,4],[3,5],[4,5],
                  [4,6]
  11  177/176   7 [1,2],[1,4],[1,6],[2,6],[2,7],[3,5],[3,6],[3,7],[4,5],
                  [4,6],[5,6]
  12  321/320   7 [1,2],[1,4],[1,5],[2,5],[2,6],[3,5],[3,6],[3,7],[4,5],
                  [4,6],[4,7],[5,6]
  13  593/592   8 [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],
                  [3,7],[3,8],[4,6],[5,8]
  14 1072/1071  9 [1,6],[1,8],[2,7],[2,8],[2,9],[3,5],[3,7],[3,9],[4,6],
                  [4,7],[4,8],[5,6],[5,8],[6,9]
  15 1856/1855  9 [1,5],[1,7],[2,5],[2,6],[2,7],[2,8],[3,6],[3,7],[3,9],
                  [4,6],[4,8],[4,9],[5,8],[5,9],[7,8]
  16 3312/3311 10 [1,7],[1,9],[2,6],[2,7],[2,8],[3,7],[3,8],[3,9],[4,5],
                  [4,6],[4,10],[5,8],[5,9],[6,9],[7,10],[8,10]

Crossrefs

Cf. A180414, A337517, A339548.

Sequence in context: A144026 A054152 A240802 * A089985 A191518 A217382

Adjacent sequences: A339805 A339806 A339807 * A339809 A339810 A339811

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Dec 18 2020

Extensions

a(18) from Hugo Pfoertner, Apr 09 2021

Status

approved