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

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, 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, 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

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

3 A198670 A198361

4 A198672 A198671

5 A198674 A198673

6 A198676 A198675

Links

Table of n, a(n) for n=1..99.

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

Cf. A198670.

Sequence in context: A067286 A256615 A201767 * A213613 A327296 A213612

Adjacent sequences: A196358 A196359 A196360 * A196362 A196363 A196364

Keyword

nonn,cons

Author

Clark Kimberling, Oct 28 2011

Status

approved