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A195445 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,2,sqrt(5) right triangle ABC.

%I A195445 #5 Mar 30 2012 18:57:45
%S A195445 6,0,1,9,5,1,6,4,4,9,0,2,1,0,9,2,7,8,2,6,3,3,7,5,8,2,4,3,3,0,3,7,4,1,
%T A195445 6,3,5,4,2,6,3,8,1,1,4,9,5,6,4,2,9,1,5,9,5,5,0,8,2,4,1,1,2,1,3,9,3,7,
%U A195445 1,4,5,5,3,6,4,4,0,6,2,4,0,7,4,7,6,8,6,1,9,7,6,4,4,6,6,2,0,5,2,9
%N A195445 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,2,sqrt(5) right triangle ABC.
%C A195445 See A195304 for definitions and a general discussion.
%e A195445 Philo(ABC,G)=0.601951644902109278263375824330374...
%t A195445 a = 1; b = 2; h = 2 a/3; k = b/3;
%t A195445 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195445 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195445 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195445 RealDigits[%, 10, 100] (* (A) A195434 *)
%t A195445 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195445 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195445 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195445 RealDigits[%, 10, 100] (* (B) A195435 *)
%t A195445 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195445 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195445 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195445 RealDigits[%, 10, 100] (* (C) A195444 *)
%t A195445 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195445 RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)
%Y A195445 Cf. A195304.
%K A195445 nonn,cons
%O A195445 0,1
%A A195445 _Clark Kimberling_, Sep 18 2011