# Initial terms and a Catalan related sequence

Creighton Dement crowdog at crowdog.de
Tue Mar 29 04:28:24 CEST 2005

```Hi all,

Taken from a recent conversation between Niel and me... concerning
something I'd found to be quite frustrating when working with FAMP.

Me:
*****
[snip}

While working with a particular floretion, a sequence shows up which I
find could be of interest. I examine it for a while and decide to submit
it. Then, anywhere from a few hours to a few weeks later, I
"re-discover" the sequence under different circumstances.  Albeit now,
the sequence has one or two extra initial terms.

Worse yet, sometimes FAMP spits out a sequence with additional terms
which don't "belong" on it (this is the case, for ex. with the more
complicated "symmetries"). When I relook up the sequence with
Superseeker, leaving off the initial term and see that the generating
function has gotten simpler, I know something's fishy.
*******

I came across a nice example of the "initial terms" problem today
(actually, it's not quite as bad as I make it sound- especially when
this doesn't happen with respect to one of my submitted sequences). To
me, it is an especially important example because it is the first time I
can remember seeing the term "Catalan" turning up in the name field of
any of my floretion sequences.

The sequence in question is  -3, -9, -19, -34, -55, -83, -119,
http://www.research.att.com/projects/OEIS?Anum=A062748
Fourth column (r=3) of FS(3) staircase array A062745.
A062745 's name is "Generalized Catalan array FS(3; n,r)".

FAMP suggests we add initial terms 1, 1, 1, 0 to the sequence A062748.
( see http://www.crowdog.de/Moonlice.html ). Superseeker reports that
the g.f. has changed to ((1-2x)(x^2-x+1))/(x-1)^4. The closed formula
then changes from a(n) = (n+1)*(n^2+8*n+18)/3!  to a(n) =
(3-n)(n^2+2)/3!

Moreover, using ves-identities we returns the relations:

2*a(n) = 3*A000124(n) - 2*A049480(n)
and
4*a(n) = 6*A000124(n) - (A004006(n) + b(n))

where the last sequence b = 2, 7, 17, 35, 64, 107, 167, 247, 350, 479,
637, 827 is unlisted and A004006 is "3-dimensional analog of centered
polygonal numbers";
A000124 is "Central polygonal numbers"

Note: it is very late so I just spot checked the above relations.

Do these terms somehow fit in with the staircase array or is a new
sequence submission unavoidable?

P.S.  A new and very neat version of FAMP will hopefully be out within
the next two days. I want to make it automatically search the OEIS
stripped.gz file (for example) and list the results within the program
FAMP itself (If you feel you are in a position to advise me on how to do
that most efficiently please do)- that part surely won't be ready in the
next two days, however.

Thanks,
Creighton

```