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A201670 Decimal expansion of least x satisfying 5*x^2 - 1 = csc(x) and 0

%I A201670 #12 Feb 07 2025 16:44:07
%S A201670 7,1,1,5,3,3,8,8,8,3,6,7,1,5,1,4,9,9,2,4,6,7,9,6,3,0,7,4,2,8,2,1,2,5,
%T A201670 5,6,7,7,2,7,9,9,2,0,5,9,3,7,5,3,9,0,7,5,8,0,6,0,8,8,5,4,9,8,9,2,8,4,
%U A201670 1,5,5,0,8,2,3,4,4,8,5,4,3,7,4,6,3,2,1,8,4,5,9,7,9,8,6,8,3,4,3
%N A201670 Decimal expansion of least x satisfying 5*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201670 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201670 G. C. Greubel, <a href="/A201670/b201670.txt">Table of n, a(n) for n = 0..10000</a>
%H A201670 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201670 least:  0.711533888367151499246796307428212556772...
%e A201670 greatest:  3.120622993086918805141273424623066002...
%t A201670 a = 5; c = -1;
%t A201670 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201670 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201670 r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
%t A201670 RealDigits[r]     (* A201670 *)
%t A201670 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201670 RealDigits[r]     (* A201671 *)
%o A201670 (PARI) a=5; c=-1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201670 Cf. A201564.
%K A201670 nonn,cons
%O A201670 0,1
%A A201670 _Clark Kimberling_, Dec 04 2011