A197519 Decimal expansion of least x>0 having cos(2*Pi*x)=(cos 2x)^2.

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, 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, 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

Offset: 0

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

			0.7507624902278812753419736314431390785682572...
		

Crossrefs

Cf. A197476.

Programs

Mathematica

b = 2 Pi; c = 2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .75, .76}, WorkingPrecision -> 200]
RealDigits[t]  (* A197519 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/2}]

Extensions

Offset corrected by Georg Fischer, Jul 28 2021