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A201583 Decimal expansion of least x satisfying 2*x^2 = csc(x) and 0

%I A201583 #12 Feb 07 2025 16:44:07
%S A201583 8,2,5,0,2,8,9,2,4,0,1,5,0,0,6,3,3,9,3,3,3,9,4,6,3,1,8,1,8,3,3,5,7,9,
%T A201583 7,8,6,9,2,2,8,7,1,0,1,4,8,2,8,7,1,0,8,8,5,2,7,4,2,5,2,0,4,2,7,7,8,5,
%U A201583 7,4,3,9,1,0,4,3,1,4,1,6,8,3,6,0,1,4,4,0,9,8,7,0,3,4,7,3,9,6,2
%N A201583 Decimal expansion of least x satisfying 2*x^2 = csc(x) and 0<x<Pi.
%C A201583 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201583 G. C. Greubel, <a href="/A201583/b201583.txt">Table of n, a(n) for n = 0..10000</a>
%H A201583 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201583 least:  0.825028924015006339333946318183357978692...
%e A201583 greatest:  3.089174211929930206560577487869973804...
%t A201583 a = 2; c = 0;
%t A201583 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201583 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201583 r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
%t A201583 RealDigits[r]    (* A201583 *)
%t A201583 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201583 RealDigits[r]    (* A201584 *)
%o A201583 (PARI) a=2; c=0; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201583 Cf. A201564.
%K A201583 nonn,cons
%O A201583 0,1
%A A201583 _Clark Kimberling_, Dec 03 2011