A198671 -id:A198671 - cp's OEIS frontend

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

Clark Kimberling, Oct 28 2011

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

			x = 1.2929430585054266652256311954691354...
min(f(x)) = -1.3155651547204494123522707...

Idris Mercer, On a function related to Chowla's cosine problem, arXiv:1206.5012v1 [math.CA], June 21 2012.
Index entries for algebraic numbers, degree 2.

Cf. A198670.

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 *)

(17+7*sqrt(7))/27 \\ Charles R Greathouse IV, Feb 07 2025

Equals (17+7*sqrt(7))/27. [Jonathan Vos Post, Jun 21 2012]

A198672 Decimal expansion of the least x>0 that gives the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x).

Original entry on oeis.org

1, 0, 0, 2, 7, 5, 0, 0, 1, 9, 2, 2, 7, 3, 0, 0, 6, 5, 9, 4, 2, 8, 3, 4, 6, 4, 8, 3, 7, 2, 3, 1, 9, 4, 3, 2, 3, 8, 6, 2, 7, 2, 3, 3, 3, 1, 8, 4, 0, 9, 7, 2, 9, 6, 4, 8, 6, 4, 9, 3, 1, 0, 1, 9, 1, 0, 5, 3, 4, 9, 1, 3, 5, 9, 2, 5, 7, 5, 1, 9, 9, 8, 6, 1, 3, 3, 4, 6, 1, 9, 7, 7, 6, 3, 6, 5, 5, 3, 7

Offset: 1

Clark Kimberling, Oct 28 2011

See A196361 for a guide to related sequences.

			x=1.002750019227300659428346483723194323862...
min(f(x))=-1.519557881642848251369426811329...

Cf. A196361, A198671.

n = 4; 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] (* A198671 *)
RealDigits[v] (* A198672 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula