This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A195471 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)=(1,sqrt(2),sqrt(3)). 5

%I

%S 6,3,5,0,7,4,3,6,8,6,2,0,6,6,8,1,3,7,5,6,2,1,5,7,6,6,1,6,4,5,4,6,4,6,

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

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

%N 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)=(1,sqrt(2),sqrt(3)).

%C See A195304 for definitions and a general discussion.

%e (A)=0.6350743686206681375621576616454646086976805000...

%t a = 1; b = Sqrt[2]; h = 2 a/3; k = b/3;

%t f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f1 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (A) A195471 *)

%t f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f2 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (B) A195472 *)

%t f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f3 = (f[t])^(1/2) /. Part[s, 1]

%t RealDigits[%, 10, 100] (* (C) A195473 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)

%Y Cf. A195304, A195471, A195472, A195473.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Sep 19 2011

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .