A307983 Consider a pentagonal bipyramid with its seven vertices P_1, ..., P_7 lying on a unit sphere; sequence gives decimal expansion of Sum_{i < j} 1/dist(P_i, P_j).

1, 4, 4, 5, 2, 9, 7, 7, 4, 1, 4, 2, 2, 1, 3, 4, 2, 9, 3, 5, 0, 4, 4, 4, 9, 1, 5, 3, 0, 6, 0, 2, 9, 2, 8, 7, 9, 0, 4, 7, 7, 8, 5, 6, 0, 5, 6, 9, 2, 5, 5, 3, 3, 6, 8, 2, 7, 1, 1, 7, 7, 7, 5, 1, 8, 0, 4, 9, 1, 3, 8, 3, 6

Offset

2,2

Comments

The seven points are the vertices of a regular pentagon inscribed in the equator plus the North and South poles.

Conjecturally, this is the solution to the Thompson problem with 7 electrons: given 7 points on the surface of a unit sphere, minimize the sum of the inverse distances between pairs of points. (This models electrostatic potential energy in the plum pudding model of the atom. N = 7 electrons is the first unresolved case.)

An octic number with denominator 2 and minimal polynomial 256x^8 - 1024x^7 - 75008x^6 + 228608x^5 + 5537120x^4 - 11456448x^3 - 103335888x^2 + 109102384x - 23637199.

Links

Table of n, a(n) for n=2..74.

Kevin S. Brown, Min-Energy Configurations of Electrons On A Sphere

Wikipedia, Thompson problem

Formula

1/2 + 10/sqrt(2) + 5/sqrt((5+sqrt(5))/2) + 5/sqrt((5-sqrt(5))/2) = 1/2 + 5*sqrt(2)*(1 + sqrt(1/2 + 1/sqrt(5))).

Example

14.452977414221342935044491530602928790477856056925533682711777518049....

Crossrefs

Sequence in context: A019922 A092171 A179778 * A330305 A011333 A283268

Adjacent sequences: A307980 A307981 A307982 * A307984 A307985 A307986

Keyword

nonn,cons

Author

Charles R Greathouse IV, May 09 2019

Status

approved