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A198677
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Decimal expansion of the absolute minimum of sin(x)+sin(2x).
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10
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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
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OFFSET
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1,2
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COMMENTS
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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)
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LINKS
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EXAMPLE
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x=5.347255851518260503318727031180159764862...
min=-1.760172593046086919405184649699273192...
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MATHEMATICA
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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]
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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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