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A201660 Decimal expansion of least x satisfying 10*x^2 = csc(x) and 0 < x < Pi.

%I A201660 #14 Feb 07 2025 16:44:07
%S A201660 4,6,9,9,3,1,6,0,6,0,0,0,5,8,8,9,2,2,8,6,8,6,5,3,5,3,5,0,6,1,8,9,1,3,
%T A201660 0,6,3,8,8,3,0,0,1,3,8,0,3,5,1,8,7,1,7,7,1,9,5,5,5,3,2,2,0,6,5,8,3,1,
%U A201660 9,3,9,2,9,8,6,4,9,6,1,7,2,5,3,0,5,5,7,6,3,7,7,6,3,2,6,7,3,4,0,8
%N A201660 Decimal expansion of least x satisfying 10*x^2 = csc(x) and 0 < x < Pi.
%C A201660 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201660 G. C. Greubel, <a href="/A201660/b201660.txt">Table of n, a(n) for n = 0..10000</a>
%H A201660 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201660 least:  0.469931606000588922868653535061891306388300...
%e A201660 greatest:  3.131394253920689935444028622238747025122...
%t A201660 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201660 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201660 r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
%t A201660 RealDigits[r]    (* A201660 *)
%t A201660 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201660 RealDigits[r]     (* A201662 *)
%o A201660 (PARI) a=10; c=0; solve(x=0.4, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201660 Cf. A201564.
%K A201660 nonn,cons
%O A201660 0,1
%A A201660 _Clark Kimberling_, Dec 04 2011
%E A201660 Terms a(90) onward corrected by _G. C. Greubel_, Sep 11 2018