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A195289
Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.
4
4, 8, 4, 7, 8, 2, 3, 8, 5, 3, 6, 6, 1, 7, 5, 3, 4, 8, 3, 3, 5, 1, 6, 5, 4, 1, 8, 0, 2, 2, 8, 1, 1, 5, 2, 7, 8, 0, 8, 8, 2, 5, 5, 4, 5, 2, 2, 8, 2, 5, 9, 9, 2, 3, 4, 1, 2, 9, 5, 4, 4, 3, 3, 4, 6, 0, 2, 1, 8, 8, 6, 9, 4, 6, 2, 9, 6, 2, 9, 3, 6, 8, 4, 9, 2, 7, 9, 5, 9, 9, 8, 0, 7, 0, 1, 2, 2, 0, 6, 2
OFFSET
0,1
COMMENTS
See A195284 for definitions and a general discussion.
EXAMPLE
Philo(ABC,I)=0.4847823853661753483351654180...
MATHEMATICA
a = 5; b = 12; c = 13;
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) A195286 *)
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) A195288 *)
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) A010487 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(A, B, C, I) A195289 *)
CROSSREFS
Cf. A195284.
Sequence in context: A155874 A160204 A200392 * A247318 A019838 A155970
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 14 2011
STATUS
approved