A195412 - OEIS

A195412

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)=(5,12,13).

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

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OFFSET

1,1

COMMENTS

See A195304 for definitions and a general discussion.

LINKS

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

EXAMPLE

(A)=3.268505398071206377100213336853865893304057...

MATHEMATICA

a = 5; b = 12; 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) A195412 *)

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) A195413 *)

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) A195414 *)

c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)

RealDigits[%, 10, 100] (* Philo(ABC, G) A195424 *)

CROSSREFS

Cf. A195413, A195414, A195424.

Sequence in context: A120232 A292961 A019444 * A069773 A122321 A072093

Adjacent sequences: A195409 A195410 A195411 * A195413 A195414 A195415

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 18 2011

STATUS

approved