A196913 - OEIS

%I #12 Apr 24 2026 12:20:14

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

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

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

%N Decimal expansion of the number x satisfying 0 < x < 2*Pi and 2x = (1 + x^2)*tan(x).

%e 0.7682171553153782504312122866979254095466915658...

%t Plot[{1/(1 + x^2), 0.874*Cos[x]}, {x, .5, 1}]

%t t = x /. FindRoot[Tan[x] == 2 x/(1 + x^2), {x, .5, 1}, WorkingPrecision -> 100]

%t RealDigits[t]    (* A196913 *)

%t c = N[Sqrt[t^4 + 6 t^2 + 1]/(t^4 + 2 t^2 + 1), 100]

%t RealDigits[c]    (* A196914 *)

%t slope = N[-c*Sin[t], 100]

%t RealDigits[slope](* A196915 *)

%o (PARI) solve(x=.5,1, tan(x)*(x^2+1)- 2*x) \\ _Charles R Greathouse IV_, Apr 24 2026

%Y Cf. A196816, A196914.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 07 2011