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A201656 Decimal expansion of least x satisfying 8*x^2 = csc(x) and 0 < x < Pi.

%I A201656 #13 Feb 07 2025 16:44:07
%S A201656 5,0,7,2,6,2,1,9,9,3,4,9,1,0,3,7,7,8,2,6,5,8,1,2,1,4,7,7,2,6,4,0,4,1,
%T A201656 9,7,6,3,8,5,8,6,3,1,5,1,4,1,8,5,6,3,4,8,1,6,0,9,3,5,5,8,2,8,6,9,5,1,
%U A201656 8,0,6,5,0,1,7,6,0,7,2,3,7,3,1,3,4,4,3,8,5,7,9,3,3,9,6,1,2,3,5
%N A201656 Decimal expansion of least x satisfying 8*x^2 = csc(x) and 0 < x < Pi.
%C A201656 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201656 G. C. Greubel, <a href="/A201656/b201656.txt">Table of n, a(n) for n = 0..10000</a>
%H A201656 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201656 least:  0.507262199349103778265812147726404197638586...
%e A201656 greatest:  3.128823571901654937275752484725028832935...
%t A201656 a = 8; c = 0;
%t A201656 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201656 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201656 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201656 RealDigits[r]   (* A201656 *)
%t A201656 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201656 RealDigits[r]   (* A201657 *)
%o A201656 (PARI) a=8; c=0; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201656 Cf. A201564.
%K A201656 nonn,cons
%O A201656 0,1
%A A201656 _Clark Kimberling_, Dec 04 2011