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A201584 Decimal expansion of greatest x satisfying 2*x^2 = csc(x) and 0

%I A201584 #12 Feb 07 2025 16:44:07
%S A201584 3,0,8,9,1,7,4,2,1,1,9,2,9,9,3,0,2,0,6,5,6,0,5,7,7,4,8,7,8,6,9,9,7,3,
%T A201584 8,0,4,9,3,7,1,6,3,0,9,6,5,6,6,7,2,1,0,0,2,6,5,8,0,5,8,8,2,2,6,9,1,1,
%U A201584 0,0,8,9,9,1,3,2,5,0,5,1,6,3,6,1,8,4,8,9,4,4,8,0,0,1,6,6,3,6,6
%N A201584 Decimal expansion of greatest x satisfying 2*x^2 = csc(x) and 0<x<Pi.
%C A201584 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201584 G. C. Greubel, <a href="/A201584/b201584.txt">Table of n, a(n) for n = 1..10000</a>
%H A201584 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201584 least:  0.825028924015006339333946318183357978692...
%e A201584 greatest:  3.089174211929930206560577487869973804...
%t A201584 a = 2; c = 0;
%t A201584 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201584 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201584 r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
%t A201584 RealDigits[r]    (* A201583 *)
%t A201584 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201584 RealDigits[r]    (* A201584 *)
%o A201584 (PARI) a=2; c=0; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201584 Cf. A201564.
%K A201584 nonn,cons
%O A201584 1,1
%A A201584 _Clark Kimberling_, Dec 03 2011