A195454 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(3),sqrt(5)).

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

Offset: 0

Author

Clark Kimberling, Sep 19 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			(A)=0.88736638964859161862798180597380875813593985...
		

Links

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

Crossrefs

Cf. A195304, A195455, A195456, A195457.

Programs

Mathematica

a = Sqrt[2]; b = Sqrt[3]; 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) A195454 *)
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) A195455 *)
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) A195456 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195457 *)