A195292 - OEIS

%I #10 Mar 30 2012 18:57:45

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

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

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

%N Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 7,24,25 right triangle ABC.

%C See A195284 for definitions and a general discussion.

%e Philo(ABC,I)=0.39368208288485419263704486771198536...

%t a = 7; b = 24; c = 25;

%t h = a (a + c)/(a + b + c); k = a*b/(a + b + c);

%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) A195290 *)

%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] (* (B)=7.5 *)

%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] (* (C) A010524 *)

%t (f1 + f2 + f3)/(a + b + c)

%t RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)

%Y Cf. A195284.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 14 2011