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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Oct 16 2011

Keywords

• nonn
• cons

Comments

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.

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.

This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			3.521337829571715698691988564454917977309181394...
		

Crossrefs

Cf. A197476.

Programs

Mathematica

b = Pi/2; c = Pi/3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 3.5, 3.53}, WorkingPrecision -> 200]
RealDigits[t] (* A197521, appears to be 3+A197383  *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 4}]
RealDigits[ 6*ArcCos[ Root[ -1 - 4# + 4#^3 & , 2]]/Pi, 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)