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 A195413 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)=(5,12,13). 5
 7, 7, 7, 7, 7, 3, 1, 7, 7, 7, 5, 1, 2, 1, 1, 5, 6, 6, 8, 6, 8, 4, 0, 3, 3, 8, 9, 2, 2, 1, 5, 4, 7, 4, 5, 8, 6, 3, 0, 2, 5, 5, 4, 4, 9, 2, 3, 1, 4, 4, 4, 0, 4, 7, 4, 0, 9, 4, 4, 8, 6, 0, 5, 7, 1, 5, 7, 9, 1, 1, 4, 8, 5, 8, 4, 2, 2, 6, 3, 9, 6, 6, 9, 8, 1, 4, 7, 1, 6, 1, 7, 5, 0, 7, 0, 6, 0, 5, 6, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See A195304 for definitions and a general discussion. LINKS EXAMPLE (B)=7.77773177751211566868403389221547... MATHEMATICA a = 5; b = 12; 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) A195412 *) 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) A195413 *) 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) A195414 *) c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c) RealDigits[%, 10, 100] (* Philo(ABC, G) A195424 *) CROSSREFS Cf. A195304. Sequence in context: A216159 A103983 A232127 * A083947 A112114 A031182 Adjacent sequences:  A195410 A195411 A195412 * A195414 A195415 A195416 KEYWORD nonn,cons AUTHOR Clark Kimberling, Sep 18 2011 STATUS approved

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