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A201565 Decimal expansion of greatest x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.

%I A201565 #20 Feb 07 2025 16:44:07
%S A201565 3,0,5,3,1,5,1,7,2,2,5,2,4,8,7,0,2,1,1,8,0,4,1,5,5,0,5,3,1,7,8,1,1,3,
%T A201565 7,4,5,9,6,2,2,4,7,6,7,8,3,9,2,0,5,5,3,4,7,5,4,1,5,4,4,1,3,9,0,6,3,7,
%U A201565 7,3,7,1,6,9,0,6,9,5,2,2,2,7,9,1,6,9,7,4,3,4,0,3,5,9,3,5,7,5,5
%N A201565 Decimal expansion of greatest x satisfying x^2 + 2 = csc(x) and 0 < x < Pi.
%C A201565 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201565 G. C. Greubel, <a href="/A201565/b201565.txt">Table of n, a(n) for n = 1..10000</a>
%H A201565 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201565 least:  0.46758094406347136736141927076686538859402537...
%e A201565 greatest:  3.05315172252487021180415505317811374596224...
%t A201565 (* Program 1: A201564, A201565 *)
%t A201565 a = 1; c = 2;
%t A201565 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201565 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201565 r = x /. FindRoot[f[x] == g[x], {x, .46, .47}, WorkingPrecision -> 110]
%t A201565 RealDigits[r]   (* A201564 *)
%t A201565 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201565 RealDigits[r]   (* A201565 *)
%t A201565 (* Program 2: implicit surface of u*x^2+v=csc(x) *)
%t A201565 f[{x_, u_, v_}] := u*x^2 + v - Csc[x];
%t A201565 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
%t A201565 ListPlot3D[Flatten[t, 1]]  (* for A201564 *)
%o A201565 (PARI) a=1; c=2; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201565 Cf. A201564.
%K A201565 nonn,cons
%O A201565 1,1
%A A201565 _Clark Kimberling_, Dec 03 2011