A197512 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x/3)^2.

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

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.

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

Examples

			3.0813735992823264623177056994554111...
		

Crossrefs

Cf. A197476.

Programs

Mathematica

b = 2; c = Pi/3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[t]   (* A197512 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 4}]