cp's OEIS Frontend

A197140 Decimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) to the line y=2x.

%I A197140 #9 Nov 08 2022 12:14:56
%S A197140 1,4,4,0,6,1,9,7,0,0,5,3,8,1,9,9,1,1,7,6,3,3,2,5,2,3,0,2,5,8,9,2,7,7,
%T A197140 4,3,5,3,7,9,9,0,9,4,7,2,6,0,8,9,0,3,3,7,7,3,9,8,4,6,7,3,6,4,2,5,6,5,
%U A197140 6,3,7,3,8,9,3,2,7,7,8,9,2,9,4,2
%N A197140 Decimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) to the line y=2x.
%C A197140 The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T.  For discussions and guides to related sequences, see A197032, A197008 and A195284.
%C A197140 A root of the polynomial 2*x^3-4*x^2+3*x-2. - _R. J. Mathar_, Nov 08 2022
%e A197140 length of Philo line:    1.6736473041529...; see A197139
%e A197140 endpoint on x axis:    (1.44062, 0)
%e A197140 endpoint on line y=2x: (0.765782, 1.53156)
%t A197140 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197140 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197140 p[t_] :=  h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 -  3 h m t^2 + m t^3 m = 2; h = 1; k = 1; (* slope m, point (h,k) *)
%t A197140 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197140 RealDigits[t]  (* A197140 *)
%t A197140 {N[t], 0} (* endpoint on x axis *)
%t A197140 {N[k*t/(k + m*t - m*h)],
%t A197140 N[m*k*t/(k + m*t - m*h)]} (* endpt. on line y=2x *)
%t A197140 d = N[Sqrt[f[t]], 100]
%t A197140 RealDigits[d] (* A197141 *)
%t A197140 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
%t A197140 ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 1.7}, AspectRatio -> Automatic]
%Y A197140 Cf. A197032, A197141, A197008, A195284.
%K A197140 nonn,cons
%O A197140 1,2
%A A197140 _Clark Kimberling_, Oct 11 2011
%E A197140 Incorrect trailing digits removed. - _R. J. Mathar_, Nov 08 2022