A195449 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,3,sqrt(10) right triangle ABC.

5, 6, 1, 7, 0, 8, 1, 6, 9, 7, 8, 3, 3, 4, 4, 5, 9, 5, 1, 7, 8, 9, 1, 5, 7, 7, 2, 9, 4, 0, 4, 7, 3, 9, 5, 6, 0, 3, 4, 0, 3, 8, 8, 0, 0, 2, 4, 5, 9, 2, 5, 6, 8, 4, 0, 2, 5, 6, 5, 9, 8, 4, 3, 8, 4, 8, 9, 1, 0, 8, 5, 4, 3, 8, 7, 7, 7, 9, 4, 2, 7, 1, 9, 7, 7, 1, 3, 4, 8, 3, 9, 8, 9, 9, 0, 9, 4, 1, 4, 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)=0.561708169783344595178915772940473956034038800...

Mathematica

a = 1; b = 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) A195446  *)
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) A195447 *)
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) A195448 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195449 *)

Crossrefs

Cf. A195304.

Sequence in context: A198728 A345144 A238135 * A020798 A374002 A021182

Adjacent sequences: A195446 A195447 A195448 * A195450 A195451 A195452

Keyword

nonn,cons

Author

Clark Kimberling, Sep 18 2011

Status

approved