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A201676 Decimal expansion of least x satisfying 8*x^2 - 1 = csc(x) and 0

%I A201676 #12 Feb 07 2025 16:44:07
%S A201676 5,9,1,0,3,8,4,5,6,3,4,1,7,9,2,3,5,6,7,5,1,1,9,5,4,8,1,8,2,5,4,6,8,7,
%T A201676 4,6,7,5,9,3,3,3,7,2,2,1,8,8,2,7,7,1,7,2,8,0,7,2,3,4,1,2,8,2,6,1,1,6,
%U A201676 7,4,3,3,0,0,3,1,5,1,9,7,1,8,0,8,7,5,5,4,1,5,4,6,9,6,5,4,3,9,9
%N A201676 Decimal expansion of least x satisfying 8*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201676 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201676 G. C. Greubel, <a href="/A201676/b201676.txt">Table of n, a(n) for n = 0..10000</a>
%H A201676 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201676 least:  0.591038456341792356751195481825468746759333...
%e A201676 greatest:  3.128657013857735929983404048440286781650...
%t A201676 a = 8; c = -1;
%t A201676 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201676 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201676 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201676 RealDigits[r]     (* A201676 *)
%t A201676 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201676 RealDigits[r]     (* A201677 *)
%o A201676 (PARI) a=8; c=-1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201676 Cf. A201564.
%K A201676 nonn,cons
%O A201676 0,1
%A A201676 _Clark Kimberling_, Dec 04 2011