A195497 - OEIS

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

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

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

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

%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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%C See A195304 for definitions and a general discussion.

%e (C)=0.862968792141037434136010433016173...

%t a = b - 1; b = 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) A195495 *)

%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) A195496 *)

%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) A195497 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)

%Y Cf. A195304.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 19 2011