A195303 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,1,sqrt(2) right triangle ABC.

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

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 * A354857 A371348 A358981

Adjacent sequences: A195300 A195301 A195302 * A195304 A195305 A195306

Keyword

nonn,cons

Author

Clark Kimberling, Sep 14 2011

Status

approved