A195496 - OEIS

A195496

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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

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

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OFFSET

1,4

COMMENTS

See A195304 for definitions and a general discussion.

LINKS

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

EXAMPLE

(B)=1.017153446754804466256798187816606336...

MATHEMATICA

a = b - 1; b = 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) A195495 *)

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) A195496 *)

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) A195497 *)

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

RealDigits[%, 10, 100] (* Philo(ABC, G) A195498 *)

CROSSREFS

Cf. A195304.

Sequence in context: A181722 A317833 A021587 * A065479 A263202 A011478

Adjacent sequences: A195493 A195494 A195495 * A195497 A195498 A195499

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 19 2011

STATUS

approved