A195300 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 28,45,53 right triangle ABC.

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

Offset

0,1

Comments

See A195284 for definitions and a general discussion.

Links

Table of n, a(n) for n=0..98.

Example

Philo(ABC,I)=0.5711409786342621686192081085873951220...

Mathematica

a = 28; b = 45; c = 53;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195298 *)
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] (* (B) A195299 *)
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] (* (C)=20*sqrt(2) *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Phil(ABC, I), A195300 *)

Crossrefs

Cf. A195284.

Sequence in context: A011378 A329346 A258716 * A019697 A217173 A246952

Adjacent sequences: A195297 A195298 A195299 * A195301 A195302 A195303

Keyword

nonn,cons

Author

Clark Kimberling, Sep 14 2011

Status

approved