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A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

%I A360030 #23 Aug 21 2025 22:18:18
%S A360030 1,3,5,8,10,11,12,14,15,16,18,19,21
%N A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.
%H A360030 IBM Research, <a href="https://research.ibm.com/haifa/ponderthis/challenges/March2025.html">Electric networks in graphs</a>, Ponder This Challenge, March 2025, asked for the only network corresponding to a(10)=15 and 4 networks for a(12)=18.
%H A360030 Hugo Pfoertner, <a href="/A360030/a360030.pdf">Illustrated examples for the terms a(6), a(7), a(8)</a>, 17 Feb 2023.
%H A360030 Hugo Pfoertner, <a href="/A360030/a360030_1.pdf">Illustrated examples for the terms a(9), a(10), a(11)</a>, 3 Apr 2023.
%H A360030 Hugo Pfoertner, <a href="/A360030/a360030_2.pdf">Illustration of a(12)=18</a>, 8 Jan 2024, showing 3 planar and 5 non-planar networks, 4 of which were required to solve the bonus question of IBM's Ponder This Challenge.
%H A360030 Hugo Pfoertner and Klaus Nagel, <a href="/A360030/a360030_3.pdf">Illustration of a(14)=21</a>, 21 Aug 2025.
%e A360030 a(2) = 1, [[1,2]]
%e A360030 .
%e A360030   1           2
%e A360030   O----R1R----O
%e A360030   R_12 = 1
%e A360030 .
%e A360030 a(3) = 3, [[1,2]^2,[2,3]]
%e A360030 .
%e A360030   1   .---R1R---.   2           3
%e A360030   O --|         |-- O ---R3R--- O
%e A360030       .---R2R---.
%e A360030 .
%e A360030   R_12 = 1/2, R_13 = 3/2,
%e A360030               R_23 = 1
%e A360030 .
%e A360030 a(4) = 5, node 5 hidden, [[1,2],[2,3]^2,[3,5],[4,5]]
%e A360030 .
%e A360030   1           2   .---R2R---.   3          (5)          4
%e A360030   O ---R1R--- O --|         |-- O ---R4R--- O ---R5R--- O
%e A360030                   .---R3R---.
%e A360030 .
%e A360030   R_12 = 1, R_13 = 3/2, R_14 = 7/2,
%e A360030             R_23 = 1/2, R_24 = 5/2,
%e A360030                         R_34 = 2
%e A360030 .
%e A360030 a(5) = 8, node 6 hidden,
%e A360030   [[1, 2], [1, 3]^2, [2, 3], [2, 4], [3, 6], [4, 5], [4, 6]]
%e A360030 .
%e A360030     1             2           4           5
%e A360030     O-----R1R-----O----R5R----O----R8R----O
%e A360030     |             |           |
%e A360030     |            R4R         R7R
%e A360030     .---R2R---.   |           |
%e A360030     |         |---O----R6R----O
%e A360030     .---R3R---.   3          (6)
%e A360030 .
%e A360030    R_12 = 5/9, R_13 = 7/18, R_14 = 19/18, R_15 = 37/18,
%e A360030                R_23 = 1/2,  R_24 = 13/18, R_25 = 31/18,
%e A360030                             R_34 =  8/9,  R_35 = 17/9,
%e A360030                                           R_45 =  1
%Y A360030 Cf. A219158, A342557, A342558, A348020.
%K A360030 nonn,hard,more,changed
%O A360030 2,2
%A A360030 _Hugo Pfoertner_ and _Rainer Rosenthal_, Feb 12 2023
%E A360030 a(14) from _Klaus Nagel_ and _Hugo Pfoertner_, Aug 21 2025