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A201654 Decimal expansion of least x satisfying 7*x^2 = csc(x) and 0 < x < Pi.

%I A201654 #13 Feb 07 2025 16:44:07
%S A201654 5,3,1,0,9,3,7,8,3,2,2,8,7,7,5,5,6,9,5,4,2,4,5,4,2,6,2,8,7,2,7,2,8,7,
%T A201654 8,8,1,2,7,0,9,7,3,8,1,6,4,0,0,6,1,0,9,0,6,3,7,8,1,0,4,1,5,3,4,8,0,6,
%U A201654 2,2,0,8,4,6,0,4,4,8,5,0,5,1,0,5,1,5,6,1,0,9,2,0,6,1,2,7,1,3,4
%N A201654 Decimal expansion of least x satisfying 7*x^2 = csc(x) and 0 < x < Pi.
%C A201654 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201654 G. C. Greubel, <a href="/A201654/b201654.txt">Table of n, a(n) for n = 0..10000</a>
%H A201654 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201654 least:  0.53109378322877556954245426287272878812709738...
%e A201654 greatest:  3.12698210171419101601393999273016371798979...
%t A201654 a = 7; c = 0;
%t A201654 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201654 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201654 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201654 RealDigits[r]   (* A201654 *)
%t A201654 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201654 RealDigits[r]   (* A201655 *)
%o A201654 (PARI) a=7; c=0; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201654 Cf. A201564.
%K A201654 nonn,cons
%O A201654 0,1
%A A201654 _Clark Kimberling_, Dec 04 2011