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A201667 Decimal expansion of greatest x satisfying 3*x^2 - 1 = csc(x) and 0

%I A201667 #12 Feb 07 2025 16:44:07
%S A201667 3,1,0,5,7,9,1,2,2,9,3,6,3,0,8,2,2,7,7,9,2,8,9,6,7,9,3,1,6,1,4,3,1,4,
%T A201667 3,0,3,5,9,5,3,6,9,7,6,5,5,5,2,9,1,7,0,3,3,2,2,8,1,2,7,8,5,1,1,4,2,9,
%U A201667 5,2,0,6,7,4,2,4,0,0,2,7,5,4,0,8,2,0,1,2,1,2,0,0,3,9,9,4,5,3,6
%N A201667 Decimal expansion of greatest x satisfying 3*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201667 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201667 G. C. Greubel, <a href="/A201667/b201667.txt">Table of n, a(n) for n = 1..10000</a>
%H A201667 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201667 least:  0.875943738724356441549462867955303876323370...
%e A201667 greatest:  3.105791229363082277928967931614314303595...
%t A201667 a = 3; c = -1;
%t A201667 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201667 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201667 r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
%t A201667 RealDigits[r]     (* A201666 *)
%t A201667 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201667 RealDigits[r]     (* A201667 *)
%o A201667 (PARI) a=3; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201667 Cf. A201564.
%K A201667 nonn,cons
%O A201667 1,1
%A A201667 _Clark Kimberling_, Dec 04 2011