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

%I A197267 #24 Feb 16 2025 08:33:15
%S A197267 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,
%T A197267 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,
%U A197267 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
%N A197267 Decimal expansion of least x>0 having sin(4x) = (sin x)^2.
%C A197267 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.
%D A197267 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.13 Calabi's triangle constant, p. 523.
%H A197267 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/CalabisTriangle.html">Calabi's Triangle</a>.
%H A197267 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A197267 0.68298269916008168797708085562538717075...
%t A197267 b = 4; c = 1; f[x_] := Sin[x]
%t A197267 t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .5, 1}, WorkingPrecision -> 100]
%t A197267 RealDigits[t] (* A197267 *)
%t A197267 Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
%t A197267 RealDigits[ ArcCos[ Root[ 1 - 3# - 4#^2 + 8#^3 &, 3]], 10, 100] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%Y A197267 Cf. A197133.
%K A197267 nonn,cons
%O A197267 0,1
%A A197267 _Clark Kimberling_, Oct 12 2011
%E A197267 Digits from a(91) on corrected by _Jean-François Alcover_, Feb 19 2013