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A201663 Decimal expansion of greatest x satisfying x^2 - 1 = csc(x) and 0

%I A201663 #13 Feb 07 2025 16:44:07
%S A201663 3,0,1,7,9,4,2,4,7,4,5,3,6,1,5,1,2,2,7,8,5,2,5,7,2,0,8,3,2,7,7,1,6,7,
%T A201663 2,5,2,8,9,4,2,8,4,3,4,1,4,3,6,2,0,0,3,3,1,9,5,6,9,9,8,3,6,0,1,0,5,7,
%U A201663 5,6,1,5,5,3,1,4,4,6,0,8,3,8,7,2,3,6,5,8,4,5,3,2,1,8,4,8,5,6,4
%N A201663 Decimal expansion of greatest x satisfying x^2 - 1 = csc(x) and 0<x<Pi.
%C A201663 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201663 G. C. Greubel, <a href="/A201663/b201663.txt">Table of n, a(n) for n = 1..10000</a>
%H A201663 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201663 least:  1.4183556185449426563353062368720819193360860...
%e A201663 greatest:  3.0179424745361512278525720832771672528942...
%t A201663 a = 1; c = -1;
%t A201663 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201663 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201663 r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
%t A201663 RealDigits[r]     (* A201661 *)
%t A201663 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201663 RealDigits[r]     (* A201663 *)
%o A201663 (PARI) a=1; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201663 Cf. A201564.
%K A201663 nonn,cons
%O A201663 1,1
%A A201663 _Clark Kimberling_, Dec 04 2011