%I #11 Apr 09 2021 05:50:46

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

%N 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.

%e a(2) = 1: 2 resistors in series produce a resistance of 2 = 1 + 1/a(1) ohm.

%e 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.

%e 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.

%e 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.

%e 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.

%e .

%e resistors vertices

%e | R | edges

%e 2 2/1 2 [1,2],[2,3]

%e 3 3/2 3 [1,2]^2,[2,3]

%e 4 4/3 4 [1,2]^3,[2,3]

%e 5 7/6 4 [1,2]^2,[2,3],[2,4],[3,4]

%e 6 11/10 5 [1,2]^2,[2,3]^2,[2,4],[3,4]

%e 7 19/18 5 [1,2]^2,[1,3],[2,3],[2,4],[3,5],[4,5]

%e 8 35/34 6 [1,2]^2,[1,3],[2,3],[2,4],[3,4],[3,5],[4,5]

%e 9 56/55 6 [1,2],[1,3],[1,4],[2,4],[3,4],[3,5],[4,5],[4,6],[5,6]

%e 10 105/104 7 [1,3],[1,4],[1,5],[2,4],[2,5],[2,6],[3,4],[3,5],[4,5],

%e [4,6]

%e 11 177/176 7 [1,2],[1,4],[1,6],[2,6],[2,7],[3,5],[3,6],[3,7],[4,5],

%e [4,6],[5,6]

%e 12 321/320 7 [1,2],[1,4],[1,5],[2,5],[2,6],[3,5],[3,6],[3,7],[4,5],

%e [4,6],[4,7],[5,6]

%e 13 593/592 8 [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],

%e [3,7],[3,8],[4,6],[5,8]

%e 14 1072/1071 9 [1,6],[1,8],[2,7],[2,8],[2,9],[3,5],[3,7],[3,9],[4,6],

%e [4,7],[4,8],[5,6],[5,8],[6,9]

%e 15 1856/1855 9 [1,5],[1,7],[2,5],[2,6],[2,7],[2,8],[3,6],[3,7],[3,9],

%e [4,6],[4,8],[4,9],[5,8],[5,9],[7,8]

%e 16 3312/3311 10 [1,7],[1,9],[2,6],[2,7],[2,8],[3,7],[3,8],[3,9],[4,5],

%e [4,6],[4,10],[5,8],[5,9],[6,9],[7,10],[8,10]

%Y Cf. A180414, A337517, A339548.

%K nonn,hard,more

%O 2,2

%A _Hugo Pfoertner_, Dec 18 2020

%E a(18) from _Hugo Pfoertner_, Apr 09 2021