A195486 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

6, 2, 0, 3, 3, 2, 2, 7, 2, 6, 5, 3, 0, 2, 5, 8, 3, 8, 0, 5, 5, 6, 8, 6, 8, 3, 7, 2, 0, 6, 0, 7, 6, 8, 8, 6, 4, 8, 3, 6, 1, 3, 4, 8, 2, 5, 4, 2, 4, 8, 1, 9, 1, 4, 6, 1, 8, 9, 3, 2, 4, 2, 5, 0, 2, 3, 7, 3, 1, 4, 7, 9, 0, 4, 8, 7, 0, 3, 3, 4, 1, 5, 9, 1, 5, 2, 4, 7, 6, 8, 7, 4, 2, 0, 1, 3, 7, 2, 0, 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)=.62033227265302583805568683720607688648361...

Mathematica

a = Sqrt[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) A195483 *)
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) A195484 *)
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) A195485 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195486 *)

Crossrefs

Cf. A195304.

Sequence in context: A224842 A318138 A130143 * A111754 A154480 A087199

Adjacent sequences: A195483 A195484 A195485 * A195487 A195488 A195489

Keyword

nonn,cons

Author

Clark Kimberling, Sep 19 2011

Status

approved