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A201677 Decimal expansion of greatest x satisfying 8*x^2 - 1 = csc(x) and 0

%I A201677 #12 Feb 07 2025 16:44:07
%S A201677 3,1,2,8,6,5,7,0,1,3,8,5,7,7,3,5,9,2,9,9,8,3,4,0,4,0,4,8,4,4,0,2,8,6,
%T A201677 7,8,1,6,5,0,0,8,6,5,6,6,6,3,7,0,4,3,3,7,2,8,4,3,8,9,4,3,9,1,0,7,2,2,
%U A201677 4,4,1,9,4,4,2,4,5,7,5,1,9,4,0,5,4,9,2,2,4,4,3,1,5,6,4,1,0,6,6
%N A201677 Decimal expansion of greatest x satisfying 8*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201677 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201677 G. C. Greubel, <a href="/A201677/b201677.txt">Table of n, a(n) for n = 1..10000</a>
%H A201677 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201677 least:  0.591038456341792356751195481825468746759333...
%e A201677 greatest:  3.128657013857735929983404048440286781650...
%t A201677 a = 8; c = -1;
%t A201677 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201677 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201677 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201677 RealDigits[r]     (* A201676 *)
%t A201677 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201677 RealDigits[r]     (* A201677 *)
%o A201677 (PARI) a=8; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201677 Cf. A201564.
%K A201677 nonn,cons
%O A201677 1,1
%A A201677 _Clark Kimberling_, Dec 04 2011