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A195303 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,1,sqrt(2) right triangle ABC. 3
6, 1, 4, 0, 5, 8, 9, 7, 1, 0, 3, 2, 2, 1, 2, 6, 1, 1, 5, 4, 6, 3, 8, 4, 8, 9, 2, 5, 3, 9, 3, 8, 5, 4, 0, 8, 2, 6, 0, 3, 6, 7, 3, 8, 6, 8, 9, 6, 9, 9, 6, 8, 9, 2, 7, 6, 4, 7, 9, 4, 1, 9, 1, 7, 6, 7, 3, 2, 8, 5, 7, 4, 5, 1, 7, 0, 3, 8, 0, 3, 8, 4, 9, 2, 8, 5, 5, 8, 3, 1, 6, 0, 3, 1, 2, 0, 5, 5, 1, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

See A195284 for definitions and a general discussion.  This constant is the maximum of Philo(ABC,I) over all triangles ABC.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

FORMULA

Equals (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))).

EXAMPLE

Philo(ABC,I)=0.614058971032212611546384892539385408260...

MATHEMATICA

a = 1; b = 1; c = Sqrt[2];

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, 1]

RealDigits[%, 10, 100] (* (A) A195301 *)

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, 4]

RealDigits[%, 10, 100] (* (B)=(A) *)

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, 1]

RealDigits[%, 10, 100] (* (C) A163960 *)

(f1 + f2 + f3)/(a + b + c)

RealDigits[%, 10, 100]  (* Philo(ABC, I), A195303 *)

PROG

(PARI) (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))) \\ Michel Marcus, Jul 27 2018

CROSSREFS

Cf. A195284.

Sequence in context: A294347 A229606 A101023 * A160199 A178646 A144540

Adjacent sequences:  A195300 A195301 A195302 * A195304 A195305 A195306

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 14 2011

STATUS

approved

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Last modified June 17 12:26 EDT 2021. Contains 345080 sequences. (Running on oeis4.)