A197519 - OEIS

%I #14 Feb 16 2025 05:30:28

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

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

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

%N Decimal expansion of least x>0 having cos(2*Pi*x)=(cos 2x)^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 This number is irrational. I cannot prove it to be algebraic or transcendental. - _Charles R Greathouse IV_, Feb 16 2025

%e 0.7507624902278812753419736314431390785682572...

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

%t t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .75, .76}, WorkingPrecision -> 200]

%t RealDigits[t]  (* A197519 *)

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

%Y Cf. A197476.

%K nonn,cons,changed

%O 0,1

%A _Clark Kimberling_, Oct 16 2011

%E Offset corrected by _Georg Fischer_, Jul 28 2021