A197509 Decimal expansion of least x > 0 having cos(2*x) = cos(2*Pi*x)^2.

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

Offset: 0

Author

Clark Kimberling, Oct 15 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

			0.40574666075124821511608477705830690...
		

Crossrefs

Cf. A197476.

Programs

Mathematica

b = 2; c = 2 Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[t] (* A197509 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/6}]