A239225 Denominator 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, 1, 1, 9, 9, 75, 225, 1225, 1225, 19845, 99225, 160083, 480249, 1288287, 3006003, 41409225, 41409225, 1329696225, 11967266025, 10667118605, 53335593025, 85530896451, 940839860961, 1371086188563, 4113258565689, 21972535073125, 285642955950625, 176021737014375

Offset

1,4

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-> denom(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, A239224.

Sequence in context: A255743 A183893 A210052 * A165830 A152018 A223744

Adjacent sequences: A239222 A239223 A239224 * A239226 A239227 A239228

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 12 2014

Status

approved