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A195300 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 28,45,53 right triangle ABC. 4
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified June 22 09:35 EDT 2021. Contains 345375 sequences. (Running on oeis4.)