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A195436 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,1,sqrt(2) right triangle ABC. 3
6, 3, 6, 2, 5, 8, 8, 2, 1, 0, 6, 1, 8, 3, 8, 3, 0, 8, 3, 9, 1, 0, 4, 9, 4, 6, 4, 7, 1, 0, 4, 3, 7, 5, 9, 8, 2, 9, 4, 2, 4, 3, 3, 0, 0, 8, 7, 6, 1, 6, 2, 8, 8, 5, 0, 0, 2, 6, 7, 6, 5, 8, 5, 1, 0, 8, 4, 8, 1, 3, 7, 7, 6, 0, 3, 6, 0, 0, 4, 4, 4, 8, 7, 7, 2, 6, 6, 5, 6, 5, 0, 1, 9, 9, 7, 7, 4, 4, 7, 3 (list; constant; graph; refs; listen; history; text; internal format)
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)=0.636258821061838308391049464710...

MATHEMATICA

a = 1; b = 1; 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) A195433 *)

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)=sqrt(8/9), -1+A179587  *)

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) A195433 *)

c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)

RealDigits[%, 10, 100] (* Philo(ABC, G) A195436 *)

CROSSREFS

Cf. A195304.

Sequence in context: A143506 A008567 A233700 * A194625 A165065 A069938

Adjacent sequences:  A195433 A195434 A195435 * A195437 A195438 A195439

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 18 2011

STATUS

approved

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Last modified October 2 04:09 EDT 2014. Contains 247527 sequences.