A197267 Decimal expansion of least x>0 having sin(4x) = (sin x)^2.

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

Offset: 0

Author

Clark Kimberling, Oct 12 2011

Keywords

• nonn
• cons

Comments

The Mathematica program includes a graph. See A197133 for a guide to least x>0 satisfying sin(b*x)=(sin(c*x))^2 for selected b and c.

Examples

			0.68298269916008168797708085562538717075...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.13 Calabi's triangle constant, p. 523.

Links

• Eric Weisstein's MathWorld, Calabi's Triangle.
• Index entries for transcendental numbers.

Crossrefs

Cf. A197133.

Programs

Mathematica

b = 4; c = 1; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .5, 1}, WorkingPrecision -> 100]
RealDigits[t] (* A197267 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
RealDigits[ ArcCos[ Root[ 1 - 3# - 4#^2 + 8#^3 &, 3]], 10, 100] // First (* Jean-François Alcover, Feb 19 2013 *)

Extensions

Digits from a(91) on corrected by Jean-François Alcover, Feb 19 2013