A195492 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
1, 0, 6, 8, 4, 7, 3, 0, 1, 7, 1, 9, 8, 6, 1, 0, 0, 3, 5, 7, 5, 3, 9, 8, 2, 0, 2, 5, 9, 9, 5, 6, 9, 6, 7, 6, 0, 9, 5, 0, 5, 7, 8, 8, 0, 8, 5, 2, 8, 7, 9, 5, 2, 7, 3, 0, 6, 8, 4, 1, 0, 7, 5, 1, 9, 4, 3, 6, 2, 1, 9, 6, 3, 2, 3, 0, 8, 7, 8, 6, 8, 7, 7, 9, 0, 6, 8, 1, 0, 2, 4, 5, 0, 6, 6, 9, 5, 6, 1, 4
Offset: 1
Examples
(B)=1.0684730171986100357539820259956...
Crossrefs
Cf. A195304.
Programs
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Mathematica
a = 1; b = Sqrt[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) A195491 *) 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) A195492 *) 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, 4] RealDigits[%, 10, 100] (* (C) A195493 *) c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c) RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)
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PARI
polrootsreal(3464*x^8 - 16028*x^7 + 40544*x^6 + 47079*x^5 - 64883*x^4 + 50115*x^3 - 43008*x^2 - 60658*x + 32292)[4] \\ Charles R Greathouse IV, Nov 26 2024
Comments