OFFSET
1,2
COMMENTS
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):
n x min(f(x))
= ======= =========
2 A140244 -9/8
LINKS
Idris Mercer, On a function related to Chowla's cosine problem, arXiv:1206.5012v1 [math.CA], June 21 2012.
FORMULA
Equals (17+7*sqrt(7))/27. [Jonathan Vos Post, Jun 21 2012]
EXAMPLE
x = 1.2929430585054266652256311954691354...
min(f(x)) = -1.3155651547204494123522707...
MATHEMATICA
n = 3; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u] (* A196361 *)
RealDigits[v] (* A198670 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
-(17 + 7*Sqrt[7])/27 // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 28 2011
STATUS
approved