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A195436 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.

%I A195436 #8 Jul 27 2018 09:36:14
%S A195436 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 A195436 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 A195436 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 A195436 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC.
%C A195436 See A195304 for definitions and a general discussion.
%H A195436 G. C. Greubel, <a href="/A195436/b195436.txt">Table of n, a(n) for n = 0..10000</a>
%e A195436 Philo(ABC,G)=0.636258821061838308391049464710...
%t A195436 a = 1; b = 1; h = 2 a/3; k = b/3;
%t A195436 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195436 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195436 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195436 RealDigits[%, 10, 100] (* (A) A195433 *)
%t A195436 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195436 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195436 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195436 RealDigits[%, 10, 100] (* (B)=sqrt(8/9), -1+A179587  *)
%t A195436 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195436 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195436 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195436 RealDigits[%, 10, 100] (* (C) A195433 *)
%t A195436 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195436 RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)
%Y A195436 Cf. A195304.
%K A195436 nonn,cons
%O A195436 0,1
%A A195436 _Clark Kimberling_, Sep 18 2011