A198679 -id:A198679 - cp's OEIS frontend

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

Clark Kimberling, Oct 29 2011

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)
...A198677...A198678
...A198679...A198728
...A198729...A198730
...A198731...A198732
...A198733...A198734

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

Cf. A198678.

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 *)

A198728 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 29 2011

See A198677 for a guide to related sequences.

			x=5.615894235655113730430510613029770525768423...
min=-2.499607604320056604368126285618043218243...

Cf. A198677.

n = 3; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198679 *)
RealDigits[v] (* A198680 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
RealDigits[3*Pi/2 + ArcSin[Root[-1 - 4x + 2x^2 + 6x^3, 3]], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)

cp's OEIS Frontend

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

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