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A201673 Decimal expansion of greatest x satisfying 6*x^2 - 1 = csc(x) and 0

%I A201673 #13 Feb 07 2025 16:44:07
%S A201673 3,1,2,4,2,1,9,9,6,2,7,0,6,0,8,1,5,9,4,8,9,8,9,0,6,2,1,0,9,2,0,2,8,5,
%T A201673 4,6,6,3,5,1,1,0,2,3,7,1,5,8,1,0,4,2,3,4,1,4,5,4,2,2,1,2,3,6,0,3,6,9,
%U A201673 3,1,5,6,3,5,3,5,6,3,8,9,6,0,0,3,7,0,8,8,0,2,3,5
%N A201673 Decimal expansion of greatest x satisfying 6*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201673 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201673 G. C. Greubel, <a href="/A201673/b201673.txt">Table of n, a(n) for n = 1..10000</a>
%H A201673 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201673 least:  0.66173655716974422624182983709400260774...
%e A201673 greatest:  3.12421996270608159489890621092028546...
%t A201673 a = 6; c = -1;
%t A201673 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201673 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201673 r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]
%t A201673 RealDigits[r]     (* A201672 *)
%t A201673 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201673 RealDigits[r]     (* A201673 *)
%o A201673 (PARI) a=6; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201673 Cf. A201564.
%K A201673 nonn,cons
%O A201673 1,1
%A A201673 _Clark Kimberling_, Dec 04 2011