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A201683 Decimal expansion of greatest x satisfying x^2 - 2 = csc(x) and 0

%I A201683 #14 Feb 07 2025 16:44:07
%S A201683 2,9,9,7,9,9,6,9,2,0,1,8,1,6,9,5,2,6,0,6,6,1,8,2,3,3,3,1,2,5,4,1,2,5,
%T A201683 8,8,7,6,5,4,9,8,3,3,6,8,1,2,0,0,3,2,4,7,4,8,8,3,6,5,9,5,1,9,3,1,0,9,
%U A201683 4,8,3,3,2,2,1,8,8,5,2,1,7,8,8,0,8,7,8,1,3,6,3,4,0,8,0,2,2,7,8
%N A201683 Decimal expansion of greatest x satisfying x^2 - 2 = csc(x) and 0<x<Pi.
%C A201683 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201683 G. C. Greubel, <a href="/A201683/b201683.txt">Table of n, a(n) for n = 1..10000</a>
%H A201683 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201683 least:  1.7360324097399950654183110774042852312772...
%e A201683 greatest:  2.9979969201816952606618233312541258876...
%t A201683 a = 1; c = -2;
%t A201683 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201683 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201683 r = x /. FindRoot[f[x] == g[x], {x, 1.7, 1.8}, WorkingPrecision -> 110]
%t A201683 RealDigits[r]     (* A201682 *)
%t A201683 r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
%t A201683 RealDigits[r]     (* A201683 *)
%o A201683 (PARI) a=1; c=-2; solve(x=2.5, 3, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201683 Cf. A201564.
%K A201683 nonn,cons
%O A201683 1,1
%A A201683 _Clark Kimberling_, Dec 04 2011