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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. 5

%I #5 Mar 30 2012 18:57:45

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

%C See A195304 for definitions and a general discussion.

%e Philo(ABC,G)=0.601951644902109278263375824330374...

%t a = 1; b = 2; 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) A195434 *)

%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) A195435 *)

%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) A195444 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)

%Y Cf. A195304.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 18 2011

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)