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A201664 Decimal expansion of least x satisfying 2*x^2 - 1 = csc(x) and 0

%I A201664 #12 Feb 07 2025 16:44:07
%S A201664 1,0,3,9,2,4,5,6,5,0,7,9,7,2,4,7,7,9,3,2,3,1,9,2,9,3,2,7,2,4,2,4,8,3,
%T A201664 7,3,0,0,0,0,8,0,9,3,7,9,8,9,5,8,9,7,9,8,3,3,6,4,4,7,1,6,0,5,2,3,5,7,
%U A201664 4,2,6,8,0,3,4,7,4,2,1,1,9,0,7,0,0,8,4,2,0,0,0,4,3,2,9,1,5,7,7
%N A201664 Decimal expansion of least x satisfying 2*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201664 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201664 G. C. Greubel, <a href="/A201664/b201664.txt">Table of n, a(n) for n = 1..10000</a>
%H A201664 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201664 least:  1.039245650797247793231929327242483730000...
%e A201664 greatest:  3.086158774377127181225948286358214524...
%t A201664 a = 2; c = -1;
%t A201664 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201664 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201664 r = x /. FindRoot[f[x] == g[x], {x, 1.0, 1.1}, WorkingPrecision -> 110]
%t A201664 RealDigits[r]     (* A201664 *)
%t A201664 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201664 RealDigits[r]      (* A201665 *)
%o A201664 (PARI) a=2; c=-1; solve(x=1, 2, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201664 Cf. A201564.
%K A201664 nonn,cons
%O A201664 1,3
%A A201664 _Clark Kimberling_, Dec 04 2011