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A201680 Decimal expansion of least x satisfying 10*x^2 - 1 = csc(x) and 0

%I A201680 #12 Feb 07 2025 16:44:07
%S A201680 5,4,2,0,6,4,4,8,2,6,8,2,1,6,0,4,8,3,7,5,5,0,4,3,1,5,2,1,6,9,4,7,6,5,
%T A201680 3,3,5,7,0,2,8,4,4,3,5,7,3,5,4,2,6,4,7,6,8,9,4,9,1,7,4,5,8,1,3,7,9,5,
%U A201680 8,0,4,9,3,7,6,5,5,6,3,1,9,0,1,5,3,8,3,4,5,9,3,2,1,4,3,7,3,5,8
%N A201680 Decimal expansion of least x satisfying 10*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201680 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201680 G. C. Greubel, <a href="/A201680/b201680.txt">Table of n, a(n) for n = 0..10000</a>
%H A201680 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201680 least:  0.54206448268216048375504315216947653357...
%e A201680 greatest:  3.13128846969356249380458505204753587...
%t A201680 a = 10; c = -1;
%t A201680 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201680 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201680 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201680 RealDigits[r]     (* A201680 *)
%t A201680 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201680 RealDigits[r]     (* A201681 *)
%o A201680 (PARI) a=10; c=-1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201680 Cf. A201564.
%K A201680 nonn,cons
%O A201680 0,1
%A A201680 _Clark Kimberling_, Dec 04 2011