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A195429 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)=(8,15,17). 5
5, 1, 7, 3, 5, 3, 1, 7, 6, 9, 8, 3, 7, 2, 5, 8, 2, 9, 8, 7, 7, 0, 6, 2, 9, 5, 8, 5, 1, 6, 9, 4, 5, 9, 7, 3, 6, 9, 1, 8, 7, 6, 6, 2, 8, 8, 3, 4, 7, 7, 9, 1, 4, 5, 8, 0, 7, 8, 6, 2, 8, 2, 2, 6, 5, 2, 8, 7, 0, 9, 0, 5, 8, 2, 6, 1, 4, 0, 1, 9, 2, 3, 8, 7, 1, 8, 8, 0, 3, 9, 1, 8, 1, 3, 2, 8, 6, 1, 8, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See A195304 for definitions and a general discussion.

LINKS

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

EXAMPLE

(A)=5.173531769837258298770629585169459736918766...

MATHEMATICA

a = 8; b = 15; 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) A195429 *)

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) A195430 *)

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) A195431 *)

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

RealDigits[%, 10, 100] (* Philo(ABC, G) A195432 *)

CROSSREFS

Cf. A195304, A195430, A195431, A195432.

Sequence in context: A051724 A084303 A011508 * A158552 A322050 A021663

Adjacent sequences:  A195426 A195427 A195428 * A195430 A195431 A195432

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 18 2011

STATUS

approved

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Last modified December 1 10:04 EST 2021. Contains 349426 sequences. (Running on oeis4.)