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A195457 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(3),sqrt(5) right triangle ABC.

%I A195457 #10 Dec 24 2017 09:26:04
%S A195457 6,3,3,0,1,2,2,8,1,0,0,0,5,8,0,7,6,6,0,0,5,5,6,0,8,8,7,9,4,6,0,6,8,1,
%T A195457 5,1,7,2,5,3,4,0,2,4,6,5,2,9,0,7,5,0,4,5,1,5,4,9,5,2,8,3,3,1,1,6,7,6,
%U A195457 6,2,4,7,2,9,7,3,8,1,6,8,8,9,2,9,3,3,2,0,4,4,9,1,9,5,7,6,0,1,1,2
%N A195457 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(3),sqrt(5) right triangle ABC.
%C A195457 See A195304 for definitions and a general discussion.
%H A195457 G. C. Greubel, <a href="/A195457/b195457.txt">Table of n, a(n) for n = 0..10000</a>
%e A195457 Philo(ABC,G)=0.6330122810005807660055608879460681...
%t A195457 a = Sqrt[2]; b = Sqrt[3]; h = 2 a/3; k = b/3;
%t A195457 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195457 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195457 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195457 RealDigits[%, 10, 100] (* (A) A195454 *)
%t A195457 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195457 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195457 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195457 RealDigits[%, 10, 100] (* (B) A195455 *)
%t A195457 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195457 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195457 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195457 RealDigits[%, 10, 100] (* (C) A195456 *)
%t A195457 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195457 RealDigits[%, 10, 100] (* Philo(ABC,G) A195457 *)
%Y A195457 Cf. A195304, A195454, A195455, A195456.
%K A195457 nonn,cons
%O A195457 0,1
%A A195457 _Clark Kimberling_, Sep 19 2011