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Decimal expansion of least x satisfying 4*x^2 = csc(x) and 0 < x < Pi.
3

%I #10 Aug 23 2018 02:15:14

%S 6,4,4,8,9,7,4,7,5,5,4,3,6,7,3,8,3,4,4,4,3,3,5,7,3,9,0,0,4,4,4,7,4,5,

%T 2,0,1,7,0,1,3,6,8,0,5,7,9,8,7,6,3,3,7,5,0,8,8,3,4,1,4,4,8,8,6,5,5,0,

%U 7,5,9,7,3,0,8,2,2,8,5,9,1,9,6,9,6,3,7,0,0,3,0,4,9,4,5,1,7,5,1

%N Decimal expansion of least x satisfying 4*x^2 = csc(x) and 0 < x < Pi.

%C See A201564 for a guide to related sequences. The Mathematica program includes a graph.

%H G. C. Greubel, <a href="/A201587/b201587.txt">Table of n, a(n) for n = 0..10000</a>

%e least: 0.6448974755436738344433573900444745201701368...

%e greatest: 3.1158390512762535211310850151952082587811...

%t a = 4; c = 0;

%t f[x_] := a*x^2 + c; g[x_] := Csc[x]

%t Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]

%t RealDigits[r] (* A201587 *)

%t r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]

%t RealDigits[r] (* A201588 *)

%o (PARI) a=4; c=0; solve(x=0.6, 0.7, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018

%Y Cf. A201564.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Dec 03 2011