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 A197521 Decimal expansion of least x>0 having cos(Pi*x/2)=(cos Pi*x/3)^2. 3

%I

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

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

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

%N Decimal expansion of least x>0 having cos(Pi*x/2)=(cos Pi*x/3)^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.

%C Conjecture: the constant here, 3.52133782..., is 3 plus the constant in A197383, the latter being the least t>0 satisfying sin(pi*t/6)=(sin pi*t/3)^2.

%e x=3.521337829571715698691988564454917977309181394...

%t b = Pi/2; c = Pi/3; f[x_] := Cos[x]

%t t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 3.5, 3.53}, WorkingPrecision -> 200]

%t RealDigits[t] (* A197521, appears to be 3+A197383 *)

%t Plot[{f[b*x], f[c*x]^2}, {x, 0, 4}]

%t RealDigits[ 6*ArcCos[ Root[ -1 - 4# + 4#^3 & , 2]]/Pi, 10, 99] // First (* _Jean-François Alcover_, Feb 19 2013 *)

%Y Cf. A197476.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 16 2011

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Last modified April 20 02:46 EDT 2019. Contains 322291 sequences. (Running on oeis4.)