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A201665 Decimal expansion of greatest x satisfying 2*x^2 - 1 = csc(x) and 0

%I A201665 #14 Feb 07 2025 16:44:07
%S A201665 3,0,8,6,1,5,8,7,7,4,3,7,7,1,2,7,1,8,1,2,2,5,9,4,8,2,8,6,3,5,8,2,1,4,
%T A201665 5,2,4,9,8,5,3,3,6,2,2,2,6,5,7,2,2,3,1,2,0,5,5,0,0,3,9,9,0,9,2,5,4,4,
%U A201665 4,8,6,8,6,5,9,4,8,8,2,3,8,2,7,4,4,0,4,8,0,4,6,9,4,9,0,8,0,2,7
%N A201665 Decimal expansion of greatest x satisfying 2*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201665 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201665 G. C. Greubel, <a href="/A201665/b201665.txt">Table of n, a(n) for n = 1..10000</a>
%H A201665 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201665 least:  1.039245650797247793231929327242483730000...
%e A201665 greatest:  3.086158774377127181225948286358214524...
%t A201665 a = 2; c = -1;
%t A201665 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201665 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201665 r = x /. FindRoot[f[x] == g[x], {x, 1.0, 1.1}, WorkingPrecision -> 110]
%t A201665 RealDigits[r]     (* A201664 *)
%t A201665 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201665 RealDigits[r]      (* A201665 *)
%o A201665 (PARI) a=2; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201665 Cf. A201564.
%K A201665 nonn,cons
%O A201665 1,1
%A A201665 _Clark Kimberling_, Dec 04 2011