A195285 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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

Offset

0,1

Comments

See A195284 for a definition of Philo(ABC,I) and general discussion.

Links

Table of n, a(n) for n=0..99.

Example

Philo(ABC,I)=0.59772335075207494572...

Mathematica

a = 3; b = 4; c = 5;
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) 195284 *)
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 A002163 *)
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) A010466 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, I) A195285 *)

Crossrefs

Cf. A195284.

Sequence in context: A344784 A220261 A365238 * A348682 A200597 A140724

Adjacent sequences: A195282 A195283 A195284 * A195286 A195287 A195288

Keyword

nonn,cons

Author

Clark Kimberling, Sep 14 2011

Status

approved