A195494 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the right triangle ABC having sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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

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)=0.631704620416679682980614446416476083...

Mathematica

a = 1; b = Sqrt[GoldenRatio]; 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) A195491 *)
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) A195492 *)
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) A195493 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195494 *)

Crossrefs

Cf. A195304.

Sequence in context: A066717 A176395 A309646 * A154969 A192741 A240264

Adjacent sequences: A195491 A195492 A195493 * A195495 A195496 A195497

Keyword

nonn,cons

Author

Clark Kimberling, Sep 19 2011

Status

approved