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A201653 Decimal expansion of greatest x satisfying 6*x^2 = csc(x) and 0 < x < Pi.

%I A201653 #15 Feb 07 2025 16:44:07
%S A201653 3,1,2,4,5,1,9,9,1,2,5,0,1,3,8,7,6,9,3,9,6,8,8,0,1,9,6,5,0,1,1,6,2,4,
%T A201653 9,9,4,1,4,4,8,7,8,6,3,8,0,3,1,2,5,4,7,4,3,5,3,6,7,5,6,7,1,9,1,1,5,1,
%U A201653 2,3,6,6,8,1,2,3,6,1,2,8,1,1,4,9,6,9,6,4,8,0,0,1,1,1,0,0,4,6,9
%N A201653 Decimal expansion of greatest x satisfying 6*x^2 = csc(x) and 0 < x < Pi.
%C A201653 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201653 G. C. Greubel, <a href="/A201653/b201653.txt">Table of n, a(n) for n = 1..10000</a>
%H A201653 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201653 least:  0.56010069491216076282384133379781207752937450...
%e A201653 greatest:  3.12451991250138769396880196501162499414487...
%t A201653 a = 6; c = 0;
%t A201653 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201653 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201653 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201653 RealDigits[r]   (* A201591 *)
%t A201653 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201653 RealDigits[r]   (* A201653 *)
%o A201653 (PARI) a=6; c=0; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201653 Cf. A201564.
%K A201653 nonn,cons
%O A201653 1,1
%A A201653 _Clark Kimberling_, Dec 03 2011