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A201666 Decimal expansion of least x satisfying 3*x^2 - 1 = csc(x) and 0

%I A201666 #12 Feb 07 2025 16:44:07
%S A201666 8,7,5,9,4,3,7,3,8,7,2,4,3,5,6,4,4,1,5,4,9,4,6,2,8,6,7,9,5,5,3,0,3,8,
%T A201666 7,6,3,2,3,3,7,0,6,0,9,4,6,0,1,1,0,6,5,5,1,5,3,7,4,4,6,4,2,5,8,2,0,8,
%U A201666 7,3,4,0,1,5,9,7,0,3,5,4,4,2,8,6,7,8,8,9,5,6,9,7,2,2,4,6,1,1,0
%N A201666 Decimal expansion of least x satisfying 3*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201666 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201666 G. C. Greubel, <a href="/A201666/b201666.txt">Table of n, a(n) for n = 0..10000</a>
%H A201666 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201666 least:  0.875943738724356441549462867955303876323370...
%e A201666 greatest:  3.105791229363082277928967931614314303595...
%t A201666 a = 3; c = -1;
%t A201666 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201666 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201666 r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
%t A201666 RealDigits[r]     (* A201666 *)
%t A201666 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201666 RealDigits[r]     (* A201667 *)
%o A201666 (PARI) a=3; c=-1; solve(x=.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201666 Cf. A201564.
%K A201666 nonn,cons
%O A201666 0,1
%A A201666 _Clark Kimberling_, Dec 04 2011