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A201681 Decimal expansion of greatest x satisfying 10*x^2 - 1 = csc(x) and 0 < x < Pi.

%I A201681 #15 Feb 07 2025 16:44:07
%S A201681 3,1,3,1,2,8,8,4,6,9,6,9,3,5,6,2,4,9,3,8,0,4,5,8,5,0,5,2,0,4,7,5,3,5,
%T A201681 8,7,7,4,0,4,4,0,0,2,4,9,2,7,1,8,5,5,6,9,0,5,3,8,6,1,2,3,0,1,6,4,4,7,
%U A201681 2,9,1,9,2,1,8,1,3,4,8,1,9,0,2,4,9,1,8,9,9,5,3
%N A201681 Decimal expansion of greatest x satisfying 10*x^2 - 1 = csc(x) and 0 < x < Pi.
%C A201681 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201681 G. C. Greubel, <a href="/A201681/b201681.txt">Table of n, a(n) for n = 1..10000</a>
%H A201681 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201681 least:  0.54206448268216048375504315216947653357...
%e A201681 greatest:  3.13128846969356249380458505204753587...
%t A201681 a = 10; c = -1;
%t A201681 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201681 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201681 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201681 RealDigits[r]     (* A201680 *)
%t A201681 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201681 RealDigits[r]     (* A201681 *)
%o A201681 (PARI) a=10; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201681 Cf. A201564.
%K A201681 nonn,cons
%O A201681 1,1
%A A201681 _Clark Kimberling_, Dec 04 2011