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

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, 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, 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

Offset: 0

Author

Clark Kimberling, Sep 19 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.6330122810005807660055608879460681...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A195304, A195454, A195455, A195456.

Programs

Mathematica

a = Sqrt[2]; b = Sqrt[3]; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195454 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B) A195455 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (C) A195456 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195457 *)