A195432 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 8,15,17 right triangle ABC.

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

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)=0.6075103745419345069024584211959403021986468...

Mathematica

a = 8; b = 15; 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) A195429 *)
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) A195430 *)
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) A195431 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195432 *)

Crossrefs

Cf. A195304.

Sequence in context: A273413 A341906 A365163 * A196915 A374407 A249651

Adjacent sequences: A195429 A195430 A195431 * A195433 A195434 A195435

Keyword

nonn,cons

Author

Clark Kimberling, Sep 18 2011

Status

approved