cp's OEIS Frontend

A195474 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(2),sqrt(3) right triangle ABC.

%I A195474 #10 Jan 27 2018 02:29:28
%S A195474 6,2,6,9,5,0,1,1,2,3,5,3,4,9,0,9,2,5,3,9,3,5,2,7,5,2,4,8,8,7,7,1,5,8,
%T A195474 9,1,9,9,9,2,6,8,6,2,7,2,9,9,8,6,9,2,3,1,1,3,4,7,5,9,8,0,7,8,6,2,3,7,
%U A195474 0,1,9,8,1,6,3,6,7,0,3,1,8,5,3,1,4,0,2,9,7,1,5,8,4,8,9,9,1,1,5,1
%N A195474 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(2),sqrt(3) right triangle ABC.
%C A195474 See A195304 for definitions and a general discussion.
%H A195474 G. C. Greubel, <a href="/A195474/b195474.txt">Table of n, a(n) for n = 0..10000</a>
%e A195474 Philo(ABC,G)=0.626950112353490925393527524887715891999...
%t A195474 a = 1; b = Sqrt[2]; h = 2 a/3; k = b/3;
%t A195474 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195474 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195474 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195474 RealDigits[%, 10, 100] (* (A) A195471 *)
%t A195474 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195474 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195474 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195474 RealDigits[%, 10, 100] (* (B) A195472 *)
%t A195474 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195474 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195474 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195474 RealDigits[%, 10, 100] (* (C) A195473 *)
%t A195474 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195474 RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)
%Y A195474 Cf. A195304.
%K A195474 nonn,cons
%O A195474 0,1
%A A195474 _Clark Kimberling_, Sep 19 2011