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Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.
3

%I #8 Jul 27 2018 09:36:14

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

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

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

%N Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.

%C See A195304 for definitions and a general discussion.

%H G. C. Greubel, <a href="/A195436/b195436.txt">Table of n, a(n) for n = 0..10000</a>

%e Philo(ABC,G)=0.636258821061838308391049464710...

%t a = 1; b = 1; 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) A195433 *)

%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)=sqrt(8/9), -1+A179587 *)

%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) A195433 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)

%Y Cf. A195304.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 18 2011