Gathering for Gardner

[Max Alekseyev]

Richard,

Your sequence is well done.

This is its o.g.f.:

x^3/(x^4 - x^3 - x^2 - x + 1)

and first 50 terms:

0, 0, 0, 1, 1, 2, 4, 6, 11, 19, 32, 56, 96, 165, 285, 490, 844, 1454,
2503, 4311, 7424, 12784, 22016, 37913, 65289, 112434, 193620, 333430,
574195, 988811, 1702816, 2932392, 5049824, 8696221, 14975621,
25789274, 44411292, 76479966, 131704911, 226806895, 390580480,
672612320, 1158294784, 1994680689, 3435007313, 5915370466,
10186763684, 17542460774, 30209587611, 52023441603

btw, what particular divisibility properties of this sequence you have in mind?

Regards,
Max

On Sun, Mar 23, 2008 at 10:48 AM, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
> So will I, but the main purpose of this message is
>  to submit a sequence, with apologies for my
>  laziness/ineptness at not using the aproved form.
>  I used to rely on members of Sloane's Dream Team
>  to take the load off Neil himself, but I believe
>  that this address now gets bumped (tho I'm trying it
>  again).
>
>  The sequence is
>
>  0, 0, 0, 1, 1, 2, 4, 6, 11, 19, 32, 56, 96, 165, 285,
>  490, 844, 1454, 2503, 4311, 7424, 12784, 22016, 37913,
>  65289, 112434, 193620,
>
>  by which time my hand calculations are suspect (I'm
>  at home, away from PARI, and have no hand calculator).
>
>  It wasn't arrived at in the heat of battle, but just
>  from the recurrence
>        a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4)
>  so that its generating function has
>               x^4 - x^3 - x^2 - x + 1
>  in its denominator  (as does the sequence
>  0,1,1,1,3,4,7,13,21,37,64,109,..., also not in OEIS?)
>
>  So it's an example of a `symmetric' quartic recurrence
>  and has some expected divisibility properties (tho not
>  as spectacular as some I've seen).
>
>  It's close to  A000786 (& A048239), A115992, A115993,
>  but there's unlikely to be any connexion).  Best,   R.
>
>
>
>  On Sun, 23 Mar 2008, N. J. A. Sloane wrote:
>
>  > Tanya,  I will be at G4G8!
>  >
>  > Neil
>
>