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A201674 Decimal expansion of least x satisfying 7*x^2 - 1 = csc(x) and 0

%I A201674 #12 Feb 07 2025 16:44:07
%S A201674 6,2,2,7,2,7,0,9,4,3,1,3,6,9,5,1,0,3,7,9,5,0,3,9,9,3,9,2,8,6,5,2,2,8,
%T A201674 9,0,1,3,8,6,1,8,3,1,8,7,7,3,8,7,6,7,8,7,6,6,7,6,5,5,3,8,3,7,6,3,8,3,
%U A201674 2,5,8,1,7,2,4,1,3,6,6,9,8,0,6,9,0,3,0,9,2,9,6,2,6,6,8,6,3,8,4
%N A201674 Decimal expansion of least x satisfying 7*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201674 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201674 G. C. Greubel, <a href="/A201674/b201674.txt">Table of n, a(n) for n = 0..10000</a>
%H A201674 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201674 least:  0.62272709431369510379503993928652289013...
%e A201674 greatest:  3.12676335481784395832471054304139350...
%t A201674 a = 7; c = -1;
%t A201674 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201674 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201674 r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]
%t A201674 RealDigits[r]     (* A201674 *)
%t A201674 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201674 RealDigits[r]     (* A201675 *)
%o A201674 (PARI) a=7; c=-1; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201674 Cf. A201564.
%K A201674 nonn,cons
%O A201674 0,1
%A A201674 _Clark Kimberling_, Dec 04 2011