A195451 - OEIS

%I #5 Mar 30 2012 18:57:45

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

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

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

%N Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(2,3,sqrt(13)).

%C See A195304 for definitions and a general discussion.

%e (B)=2.3411607930733219756148533813763834934244253886784...

%t a = 2; b = 3; h = 2 a/3; k = b/3;

%t f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f1 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (A) A195450 *)

%t f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f2 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (B) A195451 *)

%t f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f3 = (f[t])^(1/2) /. Part[s, 1]

%t RealDigits[%, 10, 100] (* (C) A195452  *)

%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195453 *)

%Y Cf. A195304.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Sep 18 2011