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A201679 Decimal expansion of greatest x satisfying 9*x^2 - 1 = csc(x) and 0

%I A201679 #12 Feb 07 2025 16:44:07
%S A201679 3,1,3,0,1,2,1,7,4,4,3,2,7,9,1,0,3,1,7,3,8,6,1,9,3,8,0,6,4,2,2,8,0,4,
%T A201679 6,4,6,8,7,3,8,2,7,3,1,0,5,6,8,6,3,4,8,6,1,1,4,2,4,1,0,1,2,3,8,5,4,7,
%U A201679 6,3,5,5,9,9,3,5,9,7,7,8,2,7,4,2,0,6,1,6,3,8,9,3,5,2,0,2,2,5,9
%N A201679 Decimal expansion of greatest x satisfying 9*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201679 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201679 G. C. Greubel, <a href="/A201679/b201679.txt">Table of n, a(n) for n = 1..10000</a>
%H A201679 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201679 least:  0.5645945233946045603454170508793526321622...
%e A201679 greatest:  3.1301217443279103173861938064228046468...
%t A201679 a = 9; c = -1;
%t A201679 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201679 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201679 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201679 RealDigits[r]     (* A201678 *)
%t A201679 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201679 RealDigits[r]     (* A201679 *)
%o A201679 (PARI) a=9; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201679 Cf. A201564.
%K A201679 nonn,cons
%O A201679 1,1
%A A201679 _Clark Kimberling_, Dec 04 2011