A195445 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,2,sqrt(5) right triangle ABC.

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, 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, 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

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)=0.601951644902109278263375824330374...

Mathematica

a = 1; b = 2; 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) A195434 *)
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) A195435 *)
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) A195444 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195445 *)

Crossrefs

Cf. A195304.

Sequence in context: A019846 A364898 A320906 * A215080 A317446 A137943

Adjacent sequences: A195442 A195443 A195444 * A195446 A195447 A195448

Keyword

nonn,cons

Author

Clark Kimberling, Sep 18 2011

Status

approved