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A195410 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).

%I A195410 #10 Jul 18 2021 10:03:44
%S A195410 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 A195410 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 A195410 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 A195410 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 A195410 See A195284 for definitions and a general discussion.
%e A195410 Philo(ABC,I)=0.4629992818729451452524915088005478716250...
%t A195410 a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
%t A195410 f = 2 a*b/(a + b + c);
%t A195410 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195410 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195410 x3 = f*Sqrt[2]
%t A195410 N[x1, 100]
%t A195410 RealDigits[%]    (* (A) A195407 *)
%t A195410 N[x2, 100]
%t A195410 RealDigits[%]    (* (B) A195408 *)
%t A195410 N[x3, 100]
%t A195410 RealDigits[%]    (* (C) A195409 *)
%t A195410 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195410 RealDigits[%]    (*  Philo(ABC,I) A195410 *)
%Y A195410 Cf. A195284.
%K A195410 nonn,cons
%O A195410 0,1
%A A195410 _Clark Kimberling_, Sep 17 2011
%E A195410 a(99) corrected by _Georg Fischer_, Jul 18 2021