A239224 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.

1, 4, 6, 32, 40, 256, 896, 4096, 4608, 65536, 360448, 524288, 1703936, 4194304, 10485760, 134217728, 142606336, 4294967296, 40802189312, 34359738368, 180388626432, 274877906944, 3161095929856, 4398046511104, 13743895347200, 70368744177664, 949978046398464

Offset

1,2

Comments

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

References

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

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

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... .

Maple

v:= proc(n) v(n):= `if`(n<3, n-1, v(n-1)/(n-2)+v(n-2)) end:
a:= n-> numer(2*n/v(n)^2):
seq(a(n), n=2..30); # Alois P. Heinz, Mar 12 2014

Mathematica

z = 40; v[1] = 0; v[2] = 1; v[n_] := v[n] = v[n - 1]/(n - 2) + v[n - 2]
u = Join[{1}, Table[2 n/v[n]^2, {n, 2, z}]];
t1 = Numerator[u]   (* A239224 *)
t2 = Denominator[u] (* A239225 *)

Crossrefs

Cf. A000796, A239225.

Sequence in context: A068720 A068402 A078250 * A087299 A229712 A331513

Adjacent sequences: A239221 A239222 A239223 * A239225 A239226 A239227

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 12 2014

Status

approved