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A201590 Decimal expansion of greatest x satisfying 5*x^2 = csc(x) and 0 < x < Pi.

%I A201590 #13 Feb 07 2025 16:44:07
%S A201590 3,1,2,1,0,5,9,4,6,3,5,2,3,8,2,7,4,1,5,3,6,0,1,7,5,7,0,0,0,3,4,0,9,2,
%T A201590 0,4,8,9,1,0,7,4,9,9,6,8,4,4,7,8,4,7,8,2,7,1,2,2,2,5,2,7,1,0,2,4,0,7,
%U A201590 1,2,3,5,0,6,2,6,9,9,8,4,0,2,3,6,0,2,1,6,0,4,6,0,7,0,9,2,7,5,4,3
%N A201590 Decimal expansion of greatest x satisfying 5*x^2 = csc(x) and 0 < x < Pi.
%C A201590 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201590 G. C. Greubel, <a href="/A201590/b201590.txt">Table of n, a(n) for n = 1..10000</a>
%H A201590 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201590 least:  0.596626819860704546761832859082141048303653100...
%e A201590 greatest:  3.121059463523827415360175700034092048910749...
%t A201590 a = 5; c = 0;
%t A201590 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201590 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201590 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201590 RealDigits[r]   (* A201589 *)
%t A201590 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201590 RealDigits[r]   (* A201590 *)
%o A201590 (PARI) a=5; c=0; solve(x=3.1, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201590 Cf. A201564.
%K A201590 nonn,cons
%O A201590 1,1
%A A201590 _Clark Kimberling_, Dec 03 2011
%E A201590 Terms a(88) onward corrected by _G. C. Greubel_, Aug 22 2018