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 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
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS See A195304 for definitions and a general discussion. LINKS EXAMPLE Philo(ABC,G)=0.575915236513482373618787369187419918767270... 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 *) CROSSREFS Cf. A195304. Sequence in context: A090987 A278813 A217167 * A065746 A065478 A109353 Adjacent sequences:  A195495 A195496 A195497 * A195499 A195500 A195501 KEYWORD nonn,cons AUTHOR Clark Kimberling, Sep 19 2011 STATUS approved

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Last modified September 22 16:50 EDT 2020. Contains 337291 sequences. (Running on oeis4.)