A195493 - OEIS

%I #5 Mar 30 2012 18:57:47

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

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

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

%N Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%C See A195304 for definitions and a general discussion.

%e (C)=0.759310778373734956811842690497767...

%t a = 1; b = Sqrt[GoldenRatio]; 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) A195491 *)

%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) A195492 *)

%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, 4]

%t RealDigits[%, 10, 100] (* (C) A195493 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)

%Y Cf. A195304.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 19 2011