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.
1, 3, 5, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21
Offset: 2
Examples
a(2) = 1, [[1,2]]
.
1 2
O----R1R----O
R_12 = 1
.
a(3) = 3, [[1,2]^2,[2,3]]
.
1 .---R1R---. 2 3
O --| |-- O ---R3R--- O
.---R2R---.
.
R_12 = 1/2, R_13 = 3/2,
R_23 = 1
.
a(4) = 5, node 5 hidden, [[1,2],[2,3]^2,[3,5],[4,5]]
.
1 2 .---R2R---. 3 (5) 4
O ---R1R--- O --| |-- O ---R4R--- O ---R5R--- O
.---R3R---.
.
R_12 = 1, R_13 = 3/2, R_14 = 7/2,
R_23 = 1/2, R_24 = 5/2,
R_34 = 2
.
a(5) = 8, node 6 hidden,
[[1, 2], [1, 3]^2, [2, 3], [2, 4], [3, 6], [4, 5], [4, 6]]
.
1 2 4 5
O-----R1R-----O----R5R----O----R8R----O
| | |
| R4R R7R
.---R2R---. | |
| |---O----R6R----O
.---R3R---. 3 (6)
.
R_12 = 5/9, R_13 = 7/18, R_14 = 19/18, R_15 = 37/18,
R_23 = 1/2, R_24 = 13/18, R_25 = 31/18,
R_34 = 8/9, R_35 = 17/9,
R_45 = 1
Links
- IBM Research, Electric networks in graphs, Ponder This Challenge, March 2025, asked for the only network corresponding to a(10)=15 and 4 networks for a(12)=18.
- Hugo Pfoertner, Illustrated examples for the terms a(6), a(7), a(8), 17 Feb 2023.
- Hugo Pfoertner, Illustrated examples for the terms a(9), a(10), a(11), 3 Apr 2023.
- Hugo Pfoertner, Illustration of a(12)=18, 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.
- Hugo Pfoertner and Klaus Nagel, Illustration of a(14)=21, 21 Aug 2025.
Extensions
a(14) from Klaus Nagel and Hugo Pfoertner, Aug 21 2025