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).

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

Offset

0,1

Comments

See A195304 for definitions and a general discussion.

Links

Table of n, a(n) for n=0..99.

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