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A195347 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,3,sqrt(10) right triangle ABC.

%I A195347 #9 Feb 11 2025 13:04:07
%S A195347 4,2,8,0,8,1,8,0,5,8,1,2,5,2,1,9,3,5,0,2,5,2,6,7,1,5,1,7,0,3,6,9,8,0,
%T A195347 9,0,1,5,6,8,4,4,3,6,5,5,7,9,1,6,1,2,6,4,4,4,1,3,4,3,5,9,8,2,0,8,3,7,
%U A195347 1,5,1,0,6,3,2,7,9,2,1,5,9,8,0,0,9,5,9,6,4,6,1,4,6,2,9,7,1,1,0,7,7
%N A195347 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,3,sqrt(10) right triangle ABC.
%C A195347 See A195284 for definitions and a general discussion.
%H A195347 <a href="/index/Al#algebraic_08">Index entries for algebraic numbers, degree 8</a>.
%e A195347 Philo(ABC,I)=0.4280818058125219350252671517036980901568443655...
%t A195347 a = 1; b = 3; c = Sqrt[10]; f = 2 a*b/(a + b + c);
%t A195347 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195347 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195347 x3 = f*Sqrt[2]
%t A195347 N[x1, 100]
%t A195347 RealDigits[%] (* A195344 *)
%t A195347 N[x2, 100]
%t A195347 RealDigits[%] (* A195345 *)
%t A195347 N[x3, 100]
%t A195347 RealDigits[%] (* A195346 *)
%t A195347 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195347 RealDigits[%] (* A195347 *)
%o A195347 (PARI) polrootsreal(6561*x^8 - 42107688*x^6 + 495305280*x^5 + 39826979224*x^4 - 60652800*x^3 - 4964068512*x^2 - 900806400*x - 44270064)[6] \\ _Charles R Greathouse IV_, Feb 11 2025
%Y A195347 Cf. A195284.
%K A195347 nonn,cons
%O A195347 0,1
%A A195347 _Clark Kimberling_, Sep 16 2011
%E A195347 a(99) corrected by _Georg Fischer_, Jul 18 2021