A197517 Decimal expansion of least x>0 having cos(Pi*x)=(cos x/2)^2.

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

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

			1.6519583136225007897646782857344472038126558395...
		

Crossrefs

Cf. A197476.

Programs

Mathematica

b = Pi; c = 1/2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.65, 1.66},
   WorkingPrecision -> 200]
RealDigits[t]  (* A197517 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 3}]