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A201682 Decimal expansion of least x satisfying x^2 - 2 = csc(x) and 0

%I A201682 #12 Feb 07 2025 16:44:07
%S A201682 1,7,3,6,0,3,2,4,0,9,7,3,9,9,9,5,0,6,5,4,1,8,3,1,1,0,7,7,4,0,4,2,8,5,
%T A201682 2,3,1,2,7,7,2,6,5,8,9,8,1,9,9,8,4,6,3,6,6,4,4,7,4,4,7,6,3,7,1,9,2,1,
%U A201682 9,4,3,1,8,7,3,3,2,6,5,0,3,8,5,1,7,0,1,7,2,1,4,3,4,4,6,9,7,1,5
%N A201682 Decimal expansion of least x satisfying x^2 - 2 = csc(x) and 0<x<Pi.
%C A201682 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201682 G. C. Greubel, <a href="/A201682/b201682.txt">Table of n, a(n) for n = 1..10000</a>
%H A201682 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201682 least:  1.7360324097399950654183110774042852312772...
%e A201682 greatest:  2.9979969201816952606618233312541258876...
%t A201682 a = 1; c = -2;
%t A201682 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201682 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201682 r = x /. FindRoot[f[x] == g[x], {x, 1.7, 1.8}, WorkingPrecision -> 110]
%t A201682 RealDigits[r]     (* A201682 *)
%t A201682 r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
%t A201682 RealDigits[r]     (* A201683 *)
%o A201682 (PARI) a=1; c=-2; solve(x=1.5, 2, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201682 Cf. A201564.
%K A201682 nonn,cons
%O A201682 1,2
%A A201682 _Clark Kimberling_, Dec 04 2011