A195492 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)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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

Offset

1,3

Comments

See A195304 for definitions and a general discussion.

Links

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

Index entries for algebraic numbers, degree 8

Example

(B)=1.0684730171986100357539820259956...

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 *)

Prog

(PARI) polrootsreal(3464*x^8 - 16028*x^7 + 40544*x^6 + 47079*x^5 - 64883*x^4 + 50115*x^3 - 43008*x^2 - 60658*x + 32292)[4] \\ Charles R Greathouse IV, Nov 26 2024

Crossrefs

Cf. A195304.

Sequence in context: A244054 A195701 A247038 * A269802 A269991 A171784

Adjacent sequences: A195489 A195490 A195491 * A195493 A195494 A195495

Keyword

nonn,cons

Author

Clark Kimberling, Sep 19 2011

Status

approved