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A201580 Decimal expansion of greatest x satisfying x^2 + 9 = csc(x) and 0 < x < Pi.

%I A201580 #13 Feb 07 2025 16:44:07
%S A201580 3,0,8,7,6,0,9,6,0,2,7,8,3,6,0,6,1,3,3,0,0,1,1,9,0,4,9,8,8,4,6,7,0,1,
%T A201580 5,0,7,7,1,5,8,0,1,9,5,4,6,6,8,5,6,6,5,3,7,9,3,4,2,4,0,9,4,1,4,2,9,0,
%U A201580 1,8,0,5,2,0,5,2,6,7,1,8,3,9,8,1,8,5,7,2,8,3,5,2,7,8,2,3,7,6,2
%N A201580 Decimal expansion of greatest x satisfying x^2 + 9 = csc(x) and 0 < x < Pi.
%C A201580 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201580 G. C. Greubel, <a href="/A201580/b201580.txt">Table of n, a(n) for n = 1..10000</a>
%H A201580 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201580 least:  0.111187649530336552411321691800657533611...
%e A201580 greatest:  3.087609602783606133001190498846701507...
%t A201580 a = 1; c = 9;
%t A201580 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201580 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201580 r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]
%t A201580 RealDigits[r]   (* A201578 *)
%t A201580 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201580 RealDigits[r]   (* A201580 *)
%o A201580 (PARI) a=1; c=9; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201580 Cf. A201564.
%K A201580 nonn,cons
%O A201580 1,1
%A A201580 _Clark Kimberling_, Dec 03 2011