A198677 Decimal expansion of the absolute minimum of sin(x)+sin(2x).

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

Offset

1,2

Comments

The function f(x)=sin(x)+sin(2x)+...+sin(nx), where n>=2, attains an absolute minimum, m, at some c between 0 and 2*pi. The absolute maximum, -m, occurs at 2*pi-c. Guide to related sequences (including graphs in Mathematica programs):

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

2....A198677...A198678

3....A198679...A198728

4....A198729...A198730

5....A198731...A198732

6....A198733...A198734

Links

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

Example

x=5.347255851518260503318727031180159764862...
min=-1.760172593046086919405184649699273192...

Mathematica

f[t_] := Sin[t]; x = Minimize[f[t] + f[2 t], t]
x = N[Minimize[f[t] + f[2 t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198677 *)
RealDigits[v]  (* A198678 *)
Plot[f[t] + f[2 t], {t, -3 Pi, 3 Pi}]
-Sqrt[3*(69 + 11*Sqrt[33])/2]/8 // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)

Crossrefs

Cf. A198678.

Sequence in context: A302201 A263171 A003299 * A154017 A100222 A196530

Adjacent sequences: A198674 A198675 A198676 * A198678 A198679 A198680

Keyword

nonn,cons

Author

Clark Kimberling, Oct 29 2011

Status

approved