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 A195491 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio). 5
 6, 2, 9, 5, 8, 1, 0, 6, 1, 3, 8, 7, 7, 1, 6, 0, 4, 4, 0, 4, 5, 4, 9, 5, 8, 7, 5, 6, 8, 8, 5, 4, 0, 6, 9, 2, 2, 3, 1, 6, 8, 4, 9, 0, 8, 3, 8, 6, 6, 0, 7, 0, 2, 9, 6, 5, 1, 1, 2, 3, 1, 3, 4, 9, 6, 2, 5, 2, 6, 6, 6, 5, 0, 5, 1, 3, 5, 9, 2, 3, 4, 6, 8, 8, 9, 9, 4, 9, 2, 9, 6, 9, 8, 9, 0, 2, 8, 7, 6, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS See A195304 for definitions and a general discussion. LINKS EXAMPLE (A)=0.62958106138771604404549587568854069... 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 *) CROSSREFS Cf. A195304, A195492, A195493, A195494. Sequence in context: A153633 A118388 A213913 * A243453 A142871 A195411 Adjacent sequences:  A195488 A195489 A195490 * A195492 A195493 A195494 KEYWORD nonn,cons AUTHOR Clark Kimberling, Sep 19 2011 STATUS approved

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