A195450 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)=(2,3,sqrt(13)).

1, 2, 7, 5, 7, 0, 6, 9, 9, 4, 4, 4, 0, 0, 5, 5, 2, 7, 6, 4, 5, 0, 3, 7, 8, 5, 5, 6, 2, 9, 1, 5, 3, 5, 2, 8, 7, 5, 2, 2, 8, 4, 4, 7, 8, 4, 4, 9, 8, 3, 3, 3, 9, 8, 7, 9, 3, 6, 7, 0, 3, 0, 2, 3, 1, 4, 9, 2, 5, 0, 0, 7, 8, 6, 0, 5, 6, 3, 7, 3, 4, 3, 6, 0, 6, 4, 1, 4, 5, 3, 9, 6, 2, 7, 5, 9, 0, 9, 2, 4

Offset

1,2

Comments

See A195304 for definitions and a general discussion.

Links

Table of n, a(n) for n=1..100.

Example

(A)=1.2757069944400552764503785562915352875228447844...

Mathematica

a = 2; b = 3; 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) A195450 *)
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) A195451 *)
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) A195452  *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195453 *)

Crossrefs

Cf. A195304, A195451, A195452, A195453.

Sequence in context: A019825 A097157 A181320 * A079833 A198737 A010593

Adjacent sequences: A195447 A195448 A195449 * A195451 A195452 A195453

Keyword

nonn,cons

Author

Clark Kimberling, Sep 18 2011

Status

approved