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A201661 Decimal expansion of least x satisfying x^2 - 1 = csc(x) and 0

%I A201661 #13 Feb 07 2025 16:44:07
%S A201661 1,4,1,8,3,5,5,6,1,8,5,4,4,9,4,2,6,5,6,3,3,5,3,0,6,2,3,6,8,7,2,0,8,1,
%T A201661 9,1,9,3,3,6,0,8,6,0,7,1,9,4,4,2,3,1,8,8,8,4,1,9,9,5,2,7,3,9,8,4,1,1,
%U A201661 0,9,3,7,8,2,6,9,7,4,6,2,0,7,9,6,9,2,0,3,5,0,8,7,4,1,3,1,5,5,1
%N A201661 Decimal expansion of least x satisfying x^2 - 1 = csc(x) and 0<x<Pi.
%C A201661 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201661 G. C. Greubel, <a href="/A201661/b201661.txt">Table of n, a(n) for n = 1..10000</a>
%H A201661 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201661 least:  1.4183556185449426563353062368720819193360860...
%e A201661 greatest:  3.0179424745361512278525720832771672528942...
%t A201661 a = 1; c = -1;
%t A201661 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201661 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201661 r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
%t A201661 RealDigits[r]     (* A201661 *)
%t A201661 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201661 RealDigits[r]     (* A201663 *)
%o A201661 (PARI) a=1; c=-1; solve(x=1, 2, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201661 Cf. A201564.
%K A201661 nonn,cons
%O A201661 1,2
%A A201661 _Clark Kimberling_, Dec 04 2011