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A201678 Decimal expansion of least x satisfying 9*x^2 - 1 = csc(x) and 0

%I A201678 #12 Feb 07 2025 16:44:07
%S A201678 5,6,4,5,9,4,5,2,3,3,9,4,6,0,4,5,6,0,3,4,5,4,1,7,0,5,0,8,7,9,3,5,2,6,
%T A201678 3,2,1,6,2,2,5,7,5,4,9,8,7,9,6,9,6,8,8,2,2,4,7,1,9,5,3,0,8,7,5,9,4,9,
%U A201678 2,5,6,3,9,7,7,8,6,7,4,0,3,4,1,3,0,6,8,3,8,6,8,7,2,9,9,0,0,2,3
%N A201678 Decimal expansion of least x satisfying 9*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201678 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201678 G. C. Greubel, <a href="/A201678/b201678.txt">Table of n, a(n) for n = 0..10000</a>
%H A201678 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201678 least:  0.5645945233946045603454170508793526321622...
%e A201678 greatest:  3.1301217443279103173861938064228046468...
%t A201678 a = 9; c = -1;
%t A201678 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201678 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201678 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201678 RealDigits[r]     (* A201678 *)
%t A201678 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201678 RealDigits[r]     (* A201679 *)
%o A201678 (PARI) a=9; c=-1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201678 Cf. A201564.
%K A201678 nonn,cons
%O A201678 0,1
%A A201678 _Clark Kimberling_, Dec 04 2011