A195471 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(2),sqrt(3)).

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

Offset

1,1

Comments

See A195304 for definitions and a general discussion.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Example

(A)=0.6350743686206681375621576616454646086976805000...

Mathematica

a = 1; b = Sqrt[2]; 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) A195471 *)
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) A195472 *)
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) A195473 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195474 *)

Crossrefs

Cf. A195304, A195471, A195472, A195473.

Sequence in context: A019150 A019165 A195490 * A305187 A065418 A228725

Adjacent sequences: A195468 A195469 A195470 * A195472 A195473 A195474

Keyword

nonn,cons

Author

Clark Kimberling, Sep 19 2011

Status

approved