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A201669 Decimal expansion of greatest x satisfying 4*x^2 - 1 = csc(x) and 0

%I A201669 #12 Feb 07 2025 16:44:07
%S A201669 3,1,1,5,1,4,6,1,1,6,0,4,0,3,6,1,2,6,7,1,5,1,9,3,1,5,4,7,4,5,0,3,2,5,
%T A201669 8,9,2,0,0,2,1,8,5,9,2,8,9,5,2,8,0,5,4,1,6,1,9,3,4,0,5,8,9,2,4,4,2,1,
%U A201669 3,9,6,5,0,1,1,7,1,2,4,8,6,6,3,9,9,7,8,0,0,3,8,5,3,4,9,5,9,9,8
%N A201669 Decimal expansion of greatest x satisfying 4*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201669 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201669 G. C. Greubel, <a href="/A201669/b201669.txt">Table of n, a(n) for n = 1..10000</a>
%H A201669 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201669 least:  0.7784767772775942312900352799867268779861...
%e A201669 greatest:  3.1151461160403612671519315474503258920...
%t A201669 a = 4; c = -1;
%t A201669 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201669 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201669 r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
%t A201669 RealDigits[r]     (* A201668 *)
%t A201669 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201669 RealDigits[r]     (* A201669 *)
%o A201669 (PARI) a=4; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201669 Cf. A201564.
%K A201669 nonn,cons
%O A201669 1,1
%A A201669 _Clark Kimberling_, Dec 04 2011