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A201668 Decimal expansion of least x satisfying 4*x^2 - 1 = csc(x) and 0

%I A201668 #12 Feb 07 2025 16:44:07
%S A201668 7,7,8,4,7,6,7,7,7,2,7,7,5,9,4,2,3,1,2,9,0,0,3,5,2,7,9,9,8,6,7,2,6,8,
%T A201668 7,7,9,8,6,1,2,4,8,6,5,6,2,6,2,4,6,1,1,5,6,8,0,0,6,2,0,9,6,5,7,7,6,3,
%U A201668 2,2,1,7,5,3,8,6,6,8,9,4,8,6,1,4,6,8,3,7,2,9,9,1,2,4,5,4,7,3,4
%N A201668 Decimal expansion of least x satisfying 4*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201668 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201668 G. C. Greubel, <a href="/A201668/b201668.txt">Table of n, a(n) for n = 0..10000</a>
%H A201668 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201668 least:  0.7784767772775942312900352799867268779861...
%e A201668 greatest:  3.1151461160403612671519315474503258920...
%t A201668 a = 4; c = -1;
%t A201668 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201668 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201668 r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
%t A201668 RealDigits[r]     (* A201668 *)
%t A201668 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201668 RealDigits[r]     (* A201669 *)
%o A201668 (PARI) a=4; c=-1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201668 Cf. A201564.
%K A201668 nonn,cons
%O A201668 0,1
%A A201668 _Clark Kimberling_, Dec 04 2011