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A196361 Decimal expansion of the absolute minimum of cos(t) + cos(2t) + cos(3t).

%I A196361 #41 Feb 07 2025 12:57:57
%S A196361 1,3,1,5,5,6,5,1,5,4,7,2,0,4,4,9,4,1,2,3,5,2,2,7,0,7,5,0,9,4,3,5,1,1,
%T A196361 9,6,2,2,2,1,1,7,8,3,0,6,7,2,5,0,7,9,6,7,6,3,9,1,7,9,0,4,1,5,3,4,8,4,
%U A196361 2,5,2,5,0,4,6,7,1,1,0,5,7,0,1,6,0,1,0,1,8,5,9,4,5,6,3,6,3,1,5
%N A196361 Decimal expansion of the absolute minimum of cos(t) + cos(2t) + cos(3t).
%C A196361 The function f(x) = cos(x) + cos(2x) + ... + cos(nx), where n >= 2, attains an absolute minimum at some c between 0 and Pi. Related sequences (with graphs in Mathematica programs):
%C A196361   n      x     min(f(x))
%C A196361   =   =======  =========
%C A196361   2   A140244    -9/8
%C A196361   3   A198670   A198361
%C A196361   4   A198672   A198671
%C A196361   5   A198674   A198673
%C A196361   6   A198676   A198675
%H A196361 Idris Mercer, <a href="http://arxiv.org/abs/1206.5012">On a function related to Chowla's cosine problem</a>, arXiv:1206.5012v1 [math.CA], June 21 2012.
%H A196361 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A196361 Equals (17+7*sqrt(7))/27. [_Jonathan Vos Post_, Jun 21 2012]
%e A196361 x = 1.2929430585054266652256311954691354...
%e A196361 min(f(x)) = -1.3155651547204494123522707...
%t A196361 n = 3; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
%t A196361 x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
%t A196361 v = 2 Pi - t /. Part[x, 2]
%t A196361 RealDigits[u]   (* A196361 *)
%t A196361 RealDigits[v]   (* A198670 *)
%t A196361 Plot[s[t], {t, -3 Pi, 3 Pi}]
%t A196361 -(17 + 7*Sqrt[7])/27 // RealDigits[#, 10, 99]& // First (* _Jean-François Alcover_, Feb 19 2013 *)
%o A196361 (PARI) (17+7*sqrt(7))/27 \\ _Charles R Greathouse IV_, Feb 07 2025
%Y A196361 Cf. A198670.
%K A196361 nonn,cons
%O A196361 1,2
%A A196361 _Clark Kimberling_, Oct 28 2011