A195486 - OEIS

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

%S 6,2,0,3,3,2,2,7,2,6,5,3,0,2,5,8,3,8,0,5,5,6,8,6,8,3,7,2,0,6,0,7,6,8,

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

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

%N Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

%C See A195304 for definitions and a general discussion.

%e Philo(ABC,G)=.62033227265302583805568683720607688648361...

%t a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)

%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) A195484 *)

%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) A195485 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)

%Y Cf. A195304.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 19 2011