A195305 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)=(3,4,5).

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

Offset

1,1

Comments

See A195304 for definitions and a general discussion.

Links

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

Example

(B)=3.288554185145030064182848108896351414361583823030...

Mathematica

a = 3; b = 4; 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) A195304 *)
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) A195305 *)
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) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]   (* Philo(ABC, G) A195411 *)

Crossrefs

Cf. A195304.

Sequence in context: A057163 A130918 A230432 * A327575 A328645 A021308

Adjacent sequences: A195302 A195303 A195304 * A195306 A195307 A195308

Keyword

nonn,cons

Author

Clark Kimberling, Sep 18 2011

Status

approved