A197511 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x/2)^2.

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

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

			0.63919919283722484044364786615341828833...
		

Crossrefs

Cf. A197476.

Programs

Mathematica

b = 2; c = Pi/2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .63, .64}, WorkingPrecision -> 110]
RealDigits[t]  (* A197511 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]

PARI

solve(x=.5,1, cos(2*x) - cos(Pi*x/2)^2) \\ Charles R Greathouse IV, Nov 26 2024

Extensions

a(98) corrected by Sean A. Irvine, Sep 08 2021