A360030 - OEIS

%I #9 Feb 18 2023 08:08:25

%S 1,3,5,8,10,11,12

%N 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 Hugo Pfoertner, <a href="/A360030/a360030.pdf">Illustrated examples for the terms a(6), a(7), a(8)</a>, 17 Feb 2023.

%e a(2) = 1, [[1,2]]

%e .

%e   1           2

%e   O----R1R----O

%e   R_12 = 1

%e .

%e a(3) = 3, [[1,2]^2,[2,3]]

%e .

%e   1   .---R1R---.   2           3

%e   O --|         |-- O ---R3R--- O

%e       .---R2R---.

%e .

%e   R_12 = 1/2, R_13 = 3/2,

%e               R_23 = 1

%e .

%e a(4) = 5, node 5 hidden, [[1,2],[2,3]^2,[3,5],[4,5]]

%e .

%e   1           2   .---R2R---.   3          (5)          4

%e   O ---R1R--- O --|         |-- O ---R4R--- O ---R5R--- O

%e                   .---R3R---.

%e .

%e   R_12 = 1, R_13 = 3/2, R_14 = 7/2,

%e             R_23 = 1/2, R_24 = 5/2,

%e                         R_34 = 2

%e .

%e a(5) = 8, node 6 hidden,

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

%e .

%e     1             2           4           5

%e     O-----R1R-----O----R5R----O----R8R----O

%e     |             |           |

%e     |            R4R         R7R

%e     .---R2R---.   |           |

%e     |         |---O----R6R----O

%e     .---R3R---.   3          (6)

%e .

%e    R_12 = 5/9, R_13 = 7/18, R_14 = 19/18, R_15 = 37/18,

%e                R_23 = 1/2,  R_24 = 13/18, R_25 = 31/18,

%e                             R_34 =  8/9,  R_35 = 17/9,

%e                                           R_45 =  1

%Y Cf. A219158, A342558, A348020.

%K nonn,more

%O 2,2

%A _Hugo Pfoertner_ and _Rainer Rosenthal_, Feb 12 2023