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A201578 Decimal expansion of least x satisfying x^2 + 10 = csc(x) and 0 < x < Pi.

%I A201578 #13 Feb 07 2025 16:44:07
%S A201578 1,0,0,0,6,6,8,8,4,0,7,2,9,1,9,3,0,9,2,7,9,8,0,5,3,8,4,4,5,9,3,8,1,1,
%T A201578 1,5,0,6,0,7,1,7,8,5,4,6,9,8,0,2,2,5,0,0,6,0,9,9,9,8,6,7,6,2,9,6,3,7,
%U A201578 6,9,7,5,0,6,9,5,8,2,6,1,7,7,6,0,5,8,1,0,9,3,5,5,1,2,5,2,5,1,2
%N A201578 Decimal expansion of least x satisfying x^2 + 10 = csc(x) and 0 < x < Pi.
%C A201578 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201578 G. C. Greubel, <a href="/A201578/b201578.txt">Table of n, a(n) for n = 0..10000</a>
%H A201578 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201578 least:  0.100066884072919309279805384459381115060...
%e A201578 greatest:  3.090421270152151453651497443899920534...
%t A201578 a = 1; c = 10;
%t A201578 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201578 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201578 r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]
%t A201578 RealDigits[r]   (* A201578 *)
%t A201578 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201578 RealDigits[r]   (* A201581 *)
%o A201578 (PARI) a=1; c=10; solve(x=0.1, 0.2, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201578 Cf. A201564.
%K A201578 nonn,cons
%O A201578 0,5
%A A201578 _Clark Kimberling_, Dec 03 2011