A239224 - OEIS

%I #21 Apr 08 2014 06:06:43

%S 1,4,6,32,40,256,896,4096,4608,65536,360448,524288,1703936,4194304,

%T 10485760,134217728,142606336,4294967296,40802189312,34359738368,

%U 180388626432,274877906944,3161095929856,4398046511104,13743895347200,70368744177664,949978046398464

%N Numerator of 2n/v(n)^2, where v(1) = 0, v(2) = 1, and v(n) = v(n-1)/(n-2) + v(n-2) for n >= 3; limit of 2n/v(n)^2 is Pi.

%C Pi = limit of A239224(n)/A239225(n), attributed to B. Cloitre in Finch.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 19.

%H Clark Kimberling, <a href="/A239224/b239224.txt">Table of n, a(n) for n = 1..1000</a>

%e Let w(n) = 2n/v(n)^2.  The first 7 values of w are 4, 6, 32/9, 40/9, 256/75, 896/225, 4096/1225, with approximations 4., 6., 3.55556, 4.44444, 3.41333, 3.98222, 3.34367; w(1000) = 3.14316..., w(10000) = 3.14175..., w(20000) = 3.14167... .

%p v:= proc(n) v(n):= `if`(n<3, n-1, v(n-1)/(n-2)+v(n-2)) end:

%p a:= n-> numer(2*n/v(n)^2):

%p seq(a(n), n=2..30); # _Alois P. Heinz_, Mar 12 2014

%t z = 40; v[1] = 0; v[2] = 1; v[n_] := v[n] = v[n - 1]/(n - 2) + v[n - 2]

%t u = Join[{1}, Table[2 n/v[n]^2, {n, 2, z}]];

%t t1 = Numerator[u]   (* A239224 *)

%t t2 = Denominator[u] (* A239225 *)

%Y Cf. A000796, A239225.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Mar 12 2014