A195482 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,sqrt(5),3 right triangle ABC.

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

Offset

0,1

Comments

See A195304 for definitions and a general discussion.

Links

Table of n, a(n) for n=0..99.

Example

Philo(ABC,G)=0.635268604839336218811506278276445852019818763...

Mathematica

a = 2; b = Sqrt[5]; 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) A195479 *)
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) A195480 *)
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) A195481 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195482 *)

Crossrefs

Cf. A195304.

Sequence in context: A228725 A286982 A153595 * A298529 A245273 A199728

Adjacent sequences: A195479 A195480 A195481 * A195483 A195484 A195485

Keyword

nonn,cons

Author

Clark Kimberling, Sep 19 2011

Status

approved