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A201658 Decimal expansion of least x satisfying 9*x^2 = csc(x) and 0 < x < Pi.

%I A201658 #11 Feb 07 2025 16:44:07
%S A201658 4,8,7,1,8,2,5,7,2,5,4,6,1,3,4,3,6,0,7,6,7,5,4,2,4,3,0,0,4,3,0,6,4,2,
%T A201658 2,0,7,8,2,6,9,9,4,2,5,6,4,3,6,3,7,9,2,2,0,6,4,9,1,8,6,8,7,1,6,1,6,4,
%U A201658 4,1,1,2,2,6,7,3,6,5,8,8,1,0,1,3,7,2,0,6,4,3,8,3,1,6,1,2,2,4,9,5
%N A201658 Decimal expansion of least x satisfying 9*x^2 = csc(x) and 0 < x < Pi.
%C A201658 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201658 G. C. Greubel, <a href="/A201658/b201658.txt">Table of n, a(n) for n = 0..10000</a>
%H A201658 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201658 least:  0.4871825725461343607675424300430642207826...
%e A201658 greatest:  3.1302527861735360350037012277754031636...
%t A201658 a = 9; c = 0;
%t A201658 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201658 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201658 r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
%t A201658 RealDigits[r]    (* A201658 *)
%t A201658 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201658 RealDigits[r]    (* A201659 *)
%o A201658 (PARI) a=9; c=0; solve(x=.4, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201658 Cf. A201564.
%K A201658 nonn,cons
%O A201658 0,1
%A A201658 _Clark Kimberling_, Dec 04 2011
%E A201658 Terms a(89) onward corrected by _G. C. Greubel_, Sep 11 2018