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A201591 Decimal expansion of least x satisfying 6*x^2 = csc(x) and 0 < x < Pi.

%I A201591 #13 Feb 07 2025 16:44:07
%S A201591 5,6,0,1,0,0,6,9,4,9,1,2,1,6,0,7,6,2,8,2,3,8,4,1,3,3,3,7,9,7,8,1,2,0,
%T A201591 7,7,5,2,9,3,7,4,5,0,3,0,3,0,8,9,6,4,1,1,5,5,8,6,1,2,2,0,4,3,0,9,0,6,
%U A201591 7,5,9,1,6,2,1,5,6,4,8,3,3,1,4,0,8,0,7,1,6,1,7,3,2,2,0,2,3,8,9,3,3
%N A201591 Decimal expansion of least x satisfying 6*x^2 = csc(x) and 0 < x < Pi.
%C A201591 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201591 G. C. Greubel, <a href="/A201591/b201591.txt">Table of n, a(n) for n = 0..10000</a>
%H A201591 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201591 least:  0.56010069491216076282384133379781207752937450...
%e A201591 greatest:  3.12451991250138769396880196501162499414487...
%t A201591 a = 6; c = 0;
%t A201591 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201591 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201591 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201591 RealDigits[r]   (* A201591 *)
%t A201591 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201591 RealDigits[r]   (* A201653 *)
%o A201591 (PARI) a=6; c=0; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201591 Cf. A201564.
%K A201591 nonn,cons
%O A201591 0,1
%A A201591 _Clark Kimberling_, Dec 03 2011
%E A201591 Terms a(90) onward corrected by _G. C. Greubel_, Aug 22 2018