A198735 Decimal expansion of the absolute minimum of f(x)+f(2x), where f(x)=sin(x)+cos(x).

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

Offset

1,1

Comments

Let f(x)=sin(x)+cos(x) and g(x)=f(x)+f(2x)+...+f(nx), where n>=2. Then f(x) attains an absolute minimum at some x between 0 and 2*pi. Guide to related sequences (including graphs in Mathematica programs):

n....x.........minimum of f(x)

2....A198735...A198736

3....A198737...A198738

4....A198739...A198740

5....A198741...A198742

6....A198743...A198744

Links

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

Example

x=4.89312267296329905539673190581...
min=-2.09298678025110342592267820137...

Mathematica

f[t_] := Sin[t] + Cos[t]
x = Minimize[f[t] + f[2 t], t]
N[x, 30]
(RealDigits[N[{#1[[1]], t /. #1[[2]]}, 110]] &)[x]
Plot[f[t] + f[2 t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
Root[27 - 162x - 207x^2 + 8x^3 + 32x^4, 1] // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)

Crossrefs

Cf. A198736.

Sequence in context: A248897 A021482 A199287 * A071120 A249417 A189963

Adjacent sequences: A198732 A198733 A198734 * A198736 A198737 A198738

Keyword

nonn,cons

Author

Clark Kimberling, Oct 29 2011

Status

approved