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A201577 Decimal expansion of greatest x satisfying x^2 + 8 = csc(x) and 0 < x < Pi.

%I A201577 #13 Feb 07 2025 16:44:07
%S A201577 3,0,8,4,4,6,4,1,4,0,5,6,4,3,8,0,8,4,9,4,5,9,1,9,0,5,9,5,3,6,4,6,4,6,
%T A201577 0,2,1,8,3,3,5,2,0,6,1,4,9,0,4,5,8,6,4,7,6,8,3,8,8,2,8,5,6,2,6,8,3,0,
%U A201577 8,4,7,2,4,3,6,7,1,4,1,4,6,2,8,5,9,3,5,3,3,4,0,8,3,2,7,8,3,5,8
%N A201577 Decimal expansion of greatest x satisfying x^2 + 8 = csc(x) and 0 < x < Pi.
%C A201577 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201577 G. C. Greubel, <a href="/A201577/b201577.txt">Table of n, a(n) for n = 1..10000</a>
%H A201577 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201577 least:  0.125081922635997441289177701653785707187...
%e A201577 greatest:  3.084464140564380849459190595364646021...
%t A201577 a = 1; c = 8;
%t A201577 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201577 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201577 r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]
%t A201577 RealDigits[r]   (* A201576 *)
%t A201577 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201577 RealDigits[r]   (* A201577 *)
%o A201577 (PARI) a=1; c=8; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201577 Cf. A201564.
%K A201577 nonn,cons
%O A201577 1,1
%A A201577 _Clark Kimberling_, Dec 03 2011