A195483 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)=(sqrt(2),sqrt(5),sqrt(7)).

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

Offset

0,1

Comments

See A195304 for definitions and a general discussion.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Index entries for algebraic numbers, degree 6.

Example

(A)=0.90534709308364721723607657678568...

Mathematica

a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)
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) A195484 *)
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) A195485 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195486 *)

Prog

(PARI) polrootsreal(5103*x^6 + 18468*x^4 + 7416*x^2 - 21296)[2] \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A195304, A195484, A195485, A195486.

Sequence in context: A382009 A247718 A154399 * A093070 A010533 A173201

Adjacent sequences: A195480 A195481 A195482 * A195484 A195485 A195486

Keyword

nonn,cons,changed

Author

Clark Kimberling, Sep 19 2011

Status

approved