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Decimal expansion of greatest x satisfying x^2 - 1 = csc(x) and 0<x<Pi.
3

%I #13 Feb 07 2025 16:44:07

%S 3,0,1,7,9,4,2,4,7,4,5,3,6,1,5,1,2,2,7,8,5,2,5,7,2,0,8,3,2,7,7,1,6,7,

%T 2,5,2,8,9,4,2,8,4,3,4,1,4,3,6,2,0,0,3,3,1,9,5,6,9,9,8,3,6,0,1,0,5,7,

%U 5,6,1,5,5,3,1,4,4,6,0,8,3,8,7,2,3,6,5,8,4,5,3,2,1,8,4,8,5,6,4

%N Decimal expansion of greatest x satisfying x^2 - 1 = csc(x) and 0<x<Pi.

%C See A201564 for a guide to related sequences. The Mathematica program includes a graph.

%H G. C. Greubel, <a href="/A201663/b201663.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e least: 1.4183556185449426563353062368720819193360860...

%e greatest: 3.0179424745361512278525720832771672528942...

%t a = 1; c = -1;

%t f[x_] := a*x^2 + c; g[x_] := Csc[x]

%t Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]

%t RealDigits[r] (* A201661 *)

%t r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]

%t RealDigits[r] (* A201663 *)

%o (PARI) a=1; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 11 2018

%Y Cf. A201564.

%K nonn,cons,changed

%O 1,1

%A _Clark Kimberling_, Dec 04 2011