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%I #19 May 09 2019 17:23:30
%S 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,
%T 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,
%U 1,3,8,3,6
%N 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).
%C The seven points are the vertices of a regular pentagon inscribed in the equator plus the North and South poles.
%C 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.)
%C 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.
%H Kevin S. Brown, <a href="https://www.mathpages.com/home/kmath005/kmath005.htm">Min-Energy Configurations of Electrons On A Sphere</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Thomson_problem">Thompson problem</a>
%F 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))).
%e 14.452977414221342935044491530602928790477856056925533682711777518049....
%K nonn,cons
%O 2,2
%A _Charles R Greathouse IV_, May 09 2019
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