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A195453 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,3,sqrt(13) right triangle ABC.

%I A195453 #5 Mar 30 2012 18:57:45
%S A195453 6,2,3,6,5,0,7,9,8,0,2,9,4,1,4,9,0,5,4,9,6,6,3,8,8,6,2,5,2,9,4,7,9,7,
%T A195453 6,9,0,5,1,3,3,9,4,3,5,5,3,4,5,7,7,0,7,0,5,1,6,0,9,6,5,2,8,9,6,5,5,7,
%U A195453 5,9,6,2,1,5,8,4,9,4,6,8,1,8,4,6,7,2,5,6,4,1,9,5,2,3,2,9,4,8,9,7
%N A195453 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,3,sqrt(13) right triangle ABC.
%C A195453 See A195304 for definitions and a general discussion.
%e A195453 Philo(ABC,G)=0.62365079802941490549663886252947976...
%t A195453 a = 2; b = 3; h = 2 a/3; k = b/3;
%t A195453 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195453 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195453 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195453 RealDigits[%, 10, 100] (* (A) A195450 *)
%t A195453 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195453 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195453 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195453 RealDigits[%, 10, 100] (* (B) A195451 *)
%t A195453 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195453 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195453 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195453 RealDigits[%, 10, 100] (* (C) A195452  *)
%t A195453 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195453 RealDigits[%, 10, 100] (* Philo(ABC,G) A195453 *)
%Y A195453 Cf. A195304.
%K A195453 nonn,cons
%O A195453 0,1
%A A195453 _Clark Kimberling_, Sep 18 2011