A195435 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)=(1,2,sqrt(5)).

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

Offset

1,2

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 6.

Example

(B)=1.387312728313820917463360240982233212...

Mathematica

a = 1; b = 2; 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) A195434 *)
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) A195435 *)
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) A195444 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195445 *)

Prog

(PARI) sqrt(subst((9*t^4 - 30*t^3 + 41*t^2 - 28*t + 8)/(9*t^2 - 12*t + 4), t, polrootsreal(27*t^3 - 54*t^2 + 36*t - 4)[1])) \\ Charles R Greathouse IV, Feb 11 2025
(PARI) polrootsreal(729*x^6 - 1215*x^4 - 297*x^2 - 125)[2] \\ Charles R Greathouse IV, Feb 11 2025

Crossrefs

Cf. A195304.

Sequence in context: A016671 A380248 A010472 * A134903 A371527 A177346

Adjacent sequences: A195432 A195433 A195434 * A195436 A195437 A195438

Keyword

nonn,cons

Author

Clark Kimberling, Sep 18 2011

Status

approved