A195474 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(2),sqrt(3) right triangle ABC.

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

Offset

0,1

Comments

See A195304 for definitions and a general discussion.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Example

Philo(ABC,G)=0.626950112353490925393527524887715891999...

Mathematica

a = 1; b = Sqrt[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) A195471 *)
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) A195472 *)
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) A195473 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195474 *)

Crossrefs

Cf. A195304.

Sequence in context: A221716 A346835 A369884 * A021945 A002371 A048595

Adjacent sequences: A195471 A195472 A195473 * A195475 A195476 A195477

Keyword

nonn,cons

Author

Clark Kimberling, Sep 19 2011

Status

approved