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

Author

Clark Kimberling, Sep 14 2011

Keywords

• nonn
• cons

Comments

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

Examples

			Philo(ABC,I)=0.614058971032212611546384892539385408260...
		

Links

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

Crossrefs

Cf. A195284.

Programs

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 *)

PARI

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

Formula

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