A195478 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(3),2 right triangle ABC.

6, 1, 3, 8, 4, 1, 7, 2, 5, 3, 9, 4, 1, 8, 6, 8, 1, 0, 6, 6, 0, 3, 6, 7, 2, 4, 6, 0, 0, 1, 3, 9, 4, 0, 2, 6, 6, 0, 7, 4, 8, 2, 7, 9, 6, 4, 8, 4, 2, 3, 9, 2, 9, 9, 9, 3, 8, 3, 0, 9, 0, 1, 7, 7, 7, 0, 9, 5, 7, 8, 7, 7, 1, 4, 1, 7, 5, 6, 4, 4, 4, 3, 6, 8, 4, 1, 2, 8, 9, 0, 4, 7, 2, 2, 2, 1, 4, 2, 9, 1

Offset: 0

Author

Clark Kimberling, Sep 19 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			Philo(ABC,G)=0.61384172539418681066036724600139402660748...
		

Crossrefs

Cf. A195304.

Programs

Mathematica

a = 1; 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) A195575 *)
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) A195576 *)
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, 4]
RealDigits[%, 10, 100] (* (C) A195577 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)