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Decimal expansion of least x>0 having cos(4x)=(cos(6x))^2.
2

%I #10 Feb 27 2013 05:12:01

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

%T 6,2,4,6,2,5,9,2,9,2,8,1,2,8,7,6,1,2,7,9,8,1,1,4,8,7,9,0,7,7,4,0,6,1,

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

%N Decimal expansion of least x>0 having cos(4x)=(cos(6x))^2.

%C The Mathematica program includes a graph. See A197476 for a guide for the least x>0 satisfying cos(b*x)=(cos(c*x))^2 for selected b and c.

%e x=0.34054490912143700309252640816391426246259292812...

%t b = 4; c = 6; f[x_] := Cos[x]

%t t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .34, .35}, WorkingPrecision -> 100]

%t RealDigits[t] (* A197485 *)

%t Plot[{f[b*x], f[c*x]^2}, {x, 0, 1/2}]

%t RealDigits[ ArcTan[ Sqrt[ Root[7#^4 - 68#^3 + 106#^2 - 68# + 7&, 1] ] ], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)

%Y Cf. A197476.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 15 2011