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Decimal expansion of the absolute minimum of sin(x)+sin(2x).
10

%I #11 Feb 19 2013 04:26:47

%S 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,

%T 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,

%U 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

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

%C 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):

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

%C 2....A198677...A198678

%C 3....A198679...A198728

%C 4....A198729...A198730

%C 5....A198731...A198732

%C 6....A198733...A198734

%e x=5.347255851518260503318727031180159764862...

%e min=-1.760172593046086919405184649699273192...

%t f[t_] := Sin[t]; x = Minimize[f[t] + f[2 t], t]

%t x = N[Minimize[f[t] + f[2 t], t], 110]; u = Part[x, 1]

%t v = t /. Part[x, 2]

%t RealDigits[u] (* A198677 *)

%t RealDigits[v] (* A198678 *)

%t Plot[f[t] + f[2 t], {t, -3 Pi, 3 Pi}]

%t -Sqrt[3*(69 + 11*Sqrt[33])/2]/8 // RealDigits[#, 10, 99]& // First (* _Jean-François Alcover_, Feb 19 2013 *)

%Y Cf. A198678.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 29 2011