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Decimal expansion of the absolute minimum of cos(t) + cos(2t) + cos(3t).
8

%I #39 Feb 15 2020 23:30:46

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

%C 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 n x min(f(x))

%C = ======= =========

%C 2 A140244 -9/8

%C 3 A198670 A198361

%C 4 A198672 A198671

%C 5 A198674 A198673

%C 6 A198676 A198675

%H 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.

%F Equals (17+7*sqrt(7))/27. [_Jonathan Vos Post_, Jun 21 2012]

%e x = 1.2929430585054266652256311954691354...

%e min(f(x)) = -1.3155651547204494123522707...

%t n = 3; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];

%t x = N[Minimize[s[t], t], 110]; u = Part[x, 1]

%t v = 2 Pi - t /. Part[x, 2]

%t RealDigits[u] (* A196361 *)

%t RealDigits[v] (* A198670 *)

%t Plot[s[t], {t, -3 Pi, 3 Pi}]

%t -(17 + 7*Sqrt[7])/27 // RealDigits[#, 10, 99]& // First (* _Jean-François Alcover_, Feb 19 2013 *)

%Y Cf. A198670.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 28 2011