A195410 - OEIS

%I #10 Jul 18 2021 10:03:44

%S 4,6,2,9,9,9,2,8,1,8,7,2,9,4,5,1,4,5,2,5,2,4,9,1,5,0,8,8,0,0,5,4,7,8,

%T 7,1,6,2,5,0,7,4,6,2,2,4,2,6,4,0,6,4,3,1,7,5,1,9,0,9,4,4,8,2,9,9,3,2,

%U 7,6,6,5,8,4,3,7,5,6,1,8,7,5,0,9,0,4,1,7,1,3,4,1,1,0,7,0,4,8,4,3,7,6

%N Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of  the right triangle ABC having sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%C See A195284 for definitions and a general discussion.

%e Philo(ABC,I)=0.4629992818729451452524915088005478716250...

%t a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];

%t f = 2 a*b/(a + b + c);

%t x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]

%t x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]

%t x3 = f*Sqrt[2]

%t N[x1, 100]

%t RealDigits[%]    (* (A) A195407 *)

%t N[x2, 100]

%t RealDigits[%]    (* (B) A195408 *)

%t N[x3, 100]

%t RealDigits[%]    (* (C) A195409 *)

%t N[(x1 + x2 + x3)/(a + b + c), 100]

%t RealDigits[%]    (*  Philo(ABC,I) A195410 *)

%Y Cf. A195284.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 17 2011

%E a(99) corrected by _Georg Fischer_, Jul 18 2021