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A201675 Decimal expansion of greatest x satisfying 7*x^2 - 1 = csc(x) and 0

%I A201675 #12 Feb 07 2025 16:44:07
%S A201675 3,1,2,6,7,6,3,3,5,4,8,1,7,8,4,3,9,5,8,3,2,4,7,1,0,5,4,3,0,4,1,3,9,3,
%T A201675 5,0,0,8,6,9,5,6,0,6,7,8,0,4,2,4,0,6,1,3,9,9,3,3,0,3,2,1,0,4,5,3,3,0,
%U A201675 3,9,5,9,0,7,3,7,1,4,3,9,0,9,5,1,1,5,5,1,5,2,7,8,9,8,4,2,3,6,0
%N A201675 Decimal expansion of greatest x satisfying 7*x^2 - 1 = csc(x) and 0<x<Pi.
%C A201675 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201675 G. C. Greubel, <a href="/A201675/b201675.txt">Table of n, a(n) for n = 1..10000</a>
%H A201675 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201675 least:  0.62272709431369510379503993928652289013...
%e A201675 greatest:  3.12676335481784395832471054304139350...
%t A201675 a = 7; c = -1;
%t A201675 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201675 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201675 r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]
%t A201675 RealDigits[r]     (* A201674 *)
%t A201675 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.14}, WorkingPrecision -> 110]
%t A201675 RealDigits[r]     (* A201675 *)
%o A201675 (PARI) a=7; c=-1; solve(x=3, 3.14, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Sep 12 2018
%Y A201675 Cf. A201564.
%K A201675 nonn,cons
%O A201675 1,1
%A A201675 _Clark Kimberling_, Dec 04 2011