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A201659 Decimal expansion of greatest x satisfying 9*x^2 = csc(x) and 0 < x < Pi.

%I A201659 #12 Feb 07 2025 16:44:07
%S A201659 3,1,3,0,2,5,2,7,8,6,1,7,3,5,3,6,0,3,5,0,0,3,7,0,1,2,2,7,7,7,5,4,0,3,
%T A201659 1,6,3,6,9,2,7,7,5,4,0,1,2,3,7,9,0,9,2,2,3,2,0,4,2,7,8,8,9,1,6,2,7,6,
%U A201659 5,5,0,4,1,7,3,6,7,9,6,3,0,5,0,2,1,9,0,5,4,6,7,0,4,3,8,2,7,8,1
%N A201659 Decimal expansion of greatest x satisfying 9*x^2 = csc(x) and 0 < x < Pi.
%C A201659 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201659 G. C. Greubel, <a href="/A201659/b201659.txt">Table of n, a(n) for n = 1..10000</a>
%H A201659 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201659 least:  0.4871825725461343607675424300430642207826...
%e A201659 greatest:  3.1302527861735360350037012277754031636...
%t A201659 a = 9; c = 0;
%t A201659 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201659 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201659 r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
%t A201659 RealDigits[r]    (* A201658 *)
%t A201659 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201659 RealDigits[r]    (* A201659 *)
%o A201659 (PARI) a=9; c=0; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018
%Y A201659 Cf. A201564.
%K A201659 nonn,cons
%O A201659 1,1
%A A201659 _Clark Kimberling_, Dec 04 2011