A195498 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the right triangle ABC having sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).
5, 7, 5, 9, 1, 5, 2, 3, 6, 5, 1, 3, 4, 8, 2, 3, 7, 3, 6, 1, 8, 7, 8, 7, 3, 6, 9, 1, 8, 7, 4, 1, 9, 9, 1, 8, 7, 6, 7, 2, 7, 0, 2, 3, 9, 6, 1, 3, 6, 8, 7, 5, 2, 7, 5, 5, 1, 8, 3, 3, 7, 7, 6, 9, 9, 0, 3, 4, 1, 9, 4, 4, 8, 1, 4, 5, 3, 5, 3, 8, 0, 7, 2, 2, 4, 9, 3, 7, 8, 8, 2, 0, 7, 2, 7, 0, 5, 4, 0, 4
Offset: 0
Examples
Philo(ABC,G)=0.575915236513482373618787369187419918767270...
Crossrefs
Cf. A195304.
Programs
-
Mathematica
a = b - 1; b = GoldenRatio; h = 2 a/3; k = b/3; 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, 4] RealDigits[%, 10, 100] (* (A) A195495 *) 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, 4] RealDigits[%, 10, 100] (* (B) A195496 *) 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, 1] RealDigits[%, 10, 100] (* (C) A195497 *) c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c) RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)
Comments