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A201657 Decimal expansion of greatest x satisfying 8*x^2 = csc(x) and 0 < x < Pi.

%I A201657 #14 Feb 07 2025 16:44:07
%S A201657 3,1,2,8,8,2,3,5,7,1,9,0,1,6,5,4,9,3,7,2,7,5,7,5,2,4,8,4,7,2,5,0,2,8,
%T A201657 8,3,2,9,3,5,6,2,6,0,4,0,3,6,8,4,2,0,1,5,6,6,1,4,6,1,4,9,2,7,1,4,3,3,
%U A201657 7,0,1,9,7,0,0,7,8,0,2,5,0,1,7,3,4,0,2,6,9,9,5,3,8,2,2,6,2,0,0
%N A201657 Decimal expansion of greatest x satisfying 8*x^2 = csc(x) and 0 < x < Pi.
%C A201657 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201657 G. C. Greubel, <a href="/A201657/b201657.txt">Table of n, a(n) for n = 1..10000</a>
%H A201657 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201657 least:  0.507262199349103778265812147726404197638586...
%e A201657 greatest:  3.128823571901654937275752484725028832935...
%t A201657 a = 8; c = 0;
%t A201657 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201657 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201657 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201657 RealDigits[r]   (* A201656 *)
%t A201657 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201657 RealDigits[r]   (* A201657 *)
%o A201657 (PARI) a=8; c=0; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 23 2018
%Y A201657 Cf. A201564.
%K A201657 nonn,cons
%O A201657 1,1
%A A201657 _Clark Kimberling_, Dec 04 2011