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A201563 Decimal expansion of greatest x satisfying x^2 + 1 = csc(x) and 0 < x < Pi.

%I A201563 #18 Feb 07 2025 16:44:07
%S A201563 3,0,4,4,0,3,0,1,6,6,1,1,9,2,3,5,2,8,1,6,2,2,9,8,3,5,4,9,8,0,1,1,0,0,
%T A201563 7,9,0,3,6,8,8,2,8,0,4,8,6,0,0,7,4,4,4,9,6,3,6,8,1,1,0,7,1,7,1,3,0,0,
%U A201563 5,4,3,7,6,0,8,6,7,4,0,3,4,7,2,7,6,9,7,1,2,4,3,8,5,2,4,3,1,5,6
%N A201563 Decimal expansion of greatest x satisfying x^2 + 1 = csc(x) and 0 < x < Pi.
%C A201563 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201563 G. C. Greubel, <a href="/A201563/b201563.txt">Table of n, a(n) for n = 1..10000</a>
%H A201563 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201563 least:  0.71942129632741031571692297003733204908...
%e A201563 greatest:  3.04403016611923528162298354980110079...
%t A201563 a = 1; c = 1;
%t A201563 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201563 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201563 r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
%t A201563 RealDigits[r]   (* A196825 *)
%t A201563 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201563 RealDigits[r]   (* A201563 *)
%o A201563 (PARI) a=1; c=1; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201563 Cf. A201564.
%K A201563 nonn,cons
%O A201563 1,1
%A A201563 _Clark Kimberling_, Dec 03 2011