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A201581 Decimal expansion of greatest x satisfying x^2 + 10 = csc(x) and 0 < x < Pi.

%I A201581 #13 Feb 07 2025 16:44:07
%S A201581 3,0,9,0,4,2,1,2,7,0,1,5,2,1,5,1,4,5,3,6,5,1,4,9,7,4,4,3,8,9,9,9,2,0,
%T A201581 5,3,4,3,8,7,8,8,2,1,3,8,3,1,5,6,3,5,0,1,4,0,8,5,5,5,5,1,8,9,9,6,6,3,
%U A201581 6,3,1,5,9,8,0,6,4,7,6,1,2,8,4,0,6,0,6,1,1,0,6,8,9,9,4,4,5,3,4
%N A201581 Decimal expansion of greatest x satisfying x^2 + 10 = csc(x) and 0 < x < Pi.
%C A201581 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201581 G. C. Greubel, <a href="/A201581/b201581.txt">Table of n, a(n) for n = 1..10000</a>
%H A201581 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201581 least:  0.100066884072919309279805384459381115060...
%e A201581 greatest:  3.090421270152151453651497443899920534...
%t A201581 a = 1; c = 10;
%t A201581 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201581 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201581 r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]
%t A201581 RealDigits[r]   (* A201578 *)
%t A201581 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201581 RealDigits[r]   (* A201581 *)
%o A201581 (PARI) a=1; c=10; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201581 Cf. A201564.
%K A201581 nonn,cons
%O A201581 1,1
%A A201581 _Clark Kimberling_, Dec 03 2011