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A201586 Decimal expansion of greatest x satisfying 3*x^2 = csc(x) and 0 < x < Pi.

%I A201586 #16 Feb 07 2025 16:44:07
%S A201586 3,1,0,7,0,5,7,0,8,4,6,6,9,2,7,9,1,3,9,4,2,1,3,3,6,3,9,7,5,8,9,0,2,3,
%T A201586 2,6,5,5,1,8,6,0,5,8,8,9,3,2,2,5,2,6,6,1,3,6,2,2,4,2,2,4,1,7,2,2,9,4,
%U A201586 1,3,6,5,7,1,6,6,3,2,5,1,8,1,3,0,2,2,1,8,8,2,7,7,7,5,6,4,8,2,6
%N A201586 Decimal expansion of greatest x satisfying 3*x^2 = csc(x) and 0 < x < Pi.
%C A201586 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201586 G. C. Greubel, <a href="/A201586/b201586.txt">Table of n, a(n) for n = 1..10000</a>
%H A201586 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201586 least:  0.71361154106545351696712348748482823114400555...
%e A201586 greatest:  3.10705708466927913942133639758902326551860...
%t A201586 a = 3; c = 0;
%t A201586 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201586 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201586 r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
%t A201586 RealDigits[r]    (* A201585 *)
%t A201586 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201586 RealDigits[r]    (* A201586 *)
%o A201586 (PARI) a=3; c=0; solve(x=3.1, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201586 Cf. A201564.
%K A201586 nonn,cons
%O A201586 1,1
%A A201586 _Clark Kimberling_, Dec 03 2011