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A383860 Central angle of the solution of the Tammes problem for 14 points on the sphere (in radians).

%I A383860 #8 May 19 2025 15:46:41
%S A383860 9,7,1,6,3,4,7,4,2,8,8,6,2,2,4,0,7,5,9,4,1,6,9,4,9,4,7,6,2,8,5,4,1,1,
%T A383860 3,8,1,7,9,0,1,0,6,8,2,7,6,8,4,7,8,2,0,7,0,2,6,8,0,3,3,4,8,1,3,5,4,5,
%U A383860 5,6,5,0,7,3,5,4,4,0,3,2,9,4,6,3,9,9,5,3,9,9,4
%N A383860 Central angle of the solution of the Tammes problem for 14 points on the sphere (in radians).
%H A383860 Laslo Hars, <a href="https://www.hars.us/Papers/Numerical_Tammes.pdf">Numerical solutions of the Tammes problem for up to 60 points</a>, Nov 2020, N=14.
%H A383860 O. R.  Musin and A. S. Tarasov, <a href="https://doi.org/10.1080/10586458.2015.1022842">The Tammes problem for N=14</a>, Exp. Math. 24 (2015) 460-468.
%H A383860 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tammes_problem">Tammes problem</a>
%e A383860 0.971634742886224075941694947628...
%p A383860 Digits := 120 ;
%p A383860 g := proc(c,x)
%p A383860     2*arccot(c*tan(x/2)) ;
%p A383860 end proc:
%p A383860 f := proc(x)
%p A383860     local c,x1,x2,x3,x4,x5 ;
%p A383860     c := cos(x)/(1-cos(x)) ;
%p A383860     x1 := Pi-x ;
%p A383860     x2 := g(c,x1) ;
%p A383860     x3 := 2*Pi-2*x-x2 ;
%p A383860     x4 := g(c,x3) ;
%p A383860     x5 := 2*Pi-x-2*x4 ;
%p A383860     2*Pi-2*x-x3-g(c,x5) ;
%p A383860 end proc:
%p A383860 x := 1.2 ; y := 1.21 ;
%p A383860 for i from 1 to 500 do
%p A383860     z := (x+y)/2 ;
%p A383860     if f(z) > 0. then
%p A383860         x := z ;
%p A383860     else
%p A383860         y := z ;
%p A383860     end if;
%p A383860     cos(z)/(1-cos(z)) ;
%p A383860     if modp(i,20) =0 then
%p A383860         arccos(%) ; evalf(%,120) ; print(%) ;
%p A383860     end if;
%p A383860     if x > y then
%p A383860         break ;
%p A383860     end if;
%p A383860 end do:
%Y A383860 Cf. A019669 (N=6), A383859 (N=7), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383861 (N=24).
%K A383860 nonn,cons
%O A383860 0,1
%A A383860 _R. J. Mathar_, May 12 2025