OFFSET
0,3
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..225
FORMULA
a(n) ~ (2*n)!*cos(1).
E.g.f. for the aerated sequence: cos(x)/(1 - x^2) = 1 + x^2/2! + 13*x^4/4! + 389*x^6/6! + ....
Recurrence equations:
a(n) = 2*n*(2*n - 1)*a(n-1) + (-1)^n with a(0) = 1.
a(n) = (4*n^2 - 2*n - 1)*a(n - 1) + (2*n - 2)*(2*n - 3)*a(n - 2) with a(0) = 1, a(1) = 1.
The latter recurrence is also satisfied by the sequence b(n) := (2*n)! with b(0) = 1, b(1) = 2. This leads to the continued fraction representation a(n) = (2*n )!*( 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/(4*n^2 - 2*n - 1) )))) ) for n >= 3. Taking the limit gives the continued fraction representation cos(1) = A049470 = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). Cf. A073743.
MAPLE
MATHEMATICA
Table[(2 n)!*Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 12}] (* Michael De Vlieger, Sep 04 2016 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Sep 02 2016
STATUS
approved