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A201671 Decimal expansion of greatest x satisfying 5*x^2 - 1 = csc(x) and 0

%I A201671 #12 Feb 07 2025 16:44:07
%S A201671 3,1,2,0,6,2,2,9,9,3,0,8,6,9,1,8,8,0,5,1,4,1,2,7,3,4,2,4,6,2,3,0,6,6,
%T A201671 0,0,2,1,7,0,0,3,5,6,2,4,5,2,7,7,9,7,3,7,1,1,0,9,8,7,8,5,9,1,2,2,8,1,
%U A201671 0,3,9,7,7,4,6,4,5,3,7,4,0,0,1,4,8,8,0,6,3,0,8,4,4,1,1,7,5,1,1
%N A201671 Decimal expansion of greatest x satisfying 5*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201671 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201671 G. C. Greubel, <a href="/A201671/b201671.txt">Table of n, a(n) for n = 1..10000</a>
%H A201671 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201671 least:  0.711533888367151499246796307428212556772...
%e A201671 greatest:  3.120622993086918805141273424623066002...
%t A201671 a = 5; c = -1;
%t A201671 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201671 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201671 r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
%t A201671 RealDigits[r]     (* A201670 *)
%t A201671 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201671 RealDigits[r]     (* A201671 *)
%o A201671 (PARI) a=5; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201671 Cf. A201564.
%K A201671 nonn,cons
%O A201671 1,1
%A A201671 _Clark Kimberling_, Dec 04 2011