A242088 Number of edges in the convex hull of a rigorous solution to Thomson's problem for n points.

0, 0, 1, 3, 6, 9, 12

Offset: 0

Author

Jonathan Sondow, May 04 2014

Keywords

• more
• hard
• nonn

Comments

Thomson’s problem is to determine the stable equilibrium configuration(s) of n particles confined to the surface of a sphere and repelling each other by an inverse square force.

Rigorous solutions are known only for n <= 6 and n = 12, with a(12) = 30.

Non-rigorous solutions are given in Wikipedia for all n <= 460. The least non-monotonic pair is 63 > 60 for n = 23 and 24, respectively.

Examples

			For n = 0 or 1 points, the convex hull is empty or a point, so there are no edges and a(0) = a(1) = 0.
		

Links

• Kevin Brown, Min-Energy Configurations of Electrons On A Sphere
• Eric Weisstein's MathWorld, Thomson Problem
• Wikipedia, Thomson problem

Crossrefs

Cf. A008486, A008585, A080865, A242617, A268487.

Formula

a(n) <= n(n-1)/2 = (n choose 2).

a(n) <= 3*n-6 = A008585(n-2) for n >= 3, since a solution to Thomson's problem gives a planar graph, which has 3*n-6 edges if it is maximal (see A008486 comments). - Jonathan Sondow, Mar 03 2018 answering a question by Joseph Wheat.