%I #19 Apr 09 2021 05:50:51
%S 2,3,4,7,11,19,35,56,105,177,321,610,1001,1893,3186,5714,10073
%N 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.
%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 1/2 2 [1,2]^2
%e 3 2/3 3 [1,2],[1,3],[2,3]
%e 4 3/4 4 [1,2],[1,4],[2,3],[3,4]
%e 5 6/7 4 [1,2]^2,[1,3],[2,4],[3,4]
%e 6 10/11 5 [1,2],[1,3],[1,4],[2,3],[3,5],[4,5]
%e 7 18/19 5 [1,2],[1,3]^2,[2,4],[3,4],[3,5],[4,5]
%e 8 34/35 6 [1,2],[1,3],[1,4],[2,5],[3,4],[3,5],[4,6],[5,6]
%e 9 55/56 6 [1,2]^2,[1,3],[2,4],[3,5],[3,6],[4,5],[4,6],[5,6]
%e 10 104/105 7 [1,4],[1,5],[2,4],[2,6],[2,7],[3,5],[3,6],[3,7],[4,6],
%e [5,7]
%e 11 176/177 7 [1,4],[1,6],[2,4],[2,5],[2,7],[3,5],[3,6],[3,7],[4,6],
%e [4,7],[5,7]
%e 12 320/321 7 [1,4],[1,6],[2,4],[2,5],[2,6],[2,7],[3,4],[3,5],[3,6],
%e [4,6],[4,7],[5,7]
%e 13 609/610 8 [1,4],[1,5],[1,7],[2,5],[2,6],[2,7],[3,4],[3,6],[3,7],
%e [4,5],[4,6],[6,8],[7,8]
%e 14 1000/1001 8 [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],
%e [3,7],[4,5],[4,6],[4,8],[6,8]
%e 15 1892/1893 9 [1,4],[1,5],[2,5],[2,6],[2,7],[2,9],[3,6],[3,7],[3,8],
%e [3,9],[4,7],[4,8],[4,9],[5,8],[6,8]
%e 16 3185/3186 9 [1,2],[1,3],[2,6],[2,7],[2,9],[3,6],[3,7],[3,8],[4,5],
%e [4,7],[4,8],[5,6],[5,8],[5,9],[6,7],[8,9]
%Y Cf. A180414, A337517, A339808.
%Y Cf. A279317, showing that maximum solutions using the square packing analogy can only be obtained for n <= 11 resistors.
%K nonn,hard,more
%O 2,1
%A _Hugo Pfoertner_, Dec 12 2020
%E a(18) from _Hugo Pfoertner_, Apr 09 2021