# [seqfan] OEIS as a combinatorics castle

Eric Angelini Eric.Angelini at kntv.be
Thu Dec 10 15:34:07 CET 2015

Hello SeqFans,

The remark hereunder is certainly not new to the list.
But I still would like to express my point of view.

This comes from the "research" I was doing this morning:

Let's add a(n) to a(n+1) and write the result as a(n+3),
leaving a "hole" for a(n+2):

S = 1, 2, __, 3, ...

We could fill this hole with anything and let S develop
itself without any more intervention. For instance, let's
decide that a(n+2)=4:

S = 1,2,4,3,6,7,9,13,16,22,29,38,51,67,89,118...

This was not in the OEIS. I quickly understood that it
was probably due to my personal choice for a(1), a(2)
and a(3). And I asked myself: What would be the "minimal"
start to illustrate this idea?

a(1)=0, a(2)=0 and a(3)=1, of course. This prompted the
Padovan seq (well, almost): https://oeis.org/A000931

I then tried the idea a(n) + a(n+1) = a(n+4), leaving
two holes:

T = 1, 2, __, __ 3, ...

And again, turned to the minimal start a(1)=0, a(2)=0,
a(3)=0 and a(4)=1. This is https://oeis.org/A017817
(well, almost).

Looking at both entries, I got disappointed (and this is
the core of my point of view): shouldn't the OEIS be
a combinatorics castle before anything else? This is,
shouldn't we _immediately_ understand, after the first
entry, or best, _in the first entry_ (the "definition")
that The Fibonacci seq, the Padovan seq and A017818 are
based on _the same idea_ witch is to add "a" to "b" and
write the result "c" somewhere in the seq itself (here,
immediately after "b", or after a single hole, or after
two holes behind "b").

My post of two days ago was going the same way:

>  I think https://oeis.org/A055265
>  and https://oeis.org/A076990 should be linked.

Zooming out: the OEIS is a wonderful tool for everyone
dealing with series of experimental (integer) results,
as we know. But for me, this should be a side-effect,
not the main goal! The main goal being to _construct_
sequences, not _collect_ them in the "nature". In other
words (but you get the point) to do "pure" maths and
not "applied" ones.

I see two caveats:
1) a single seq as the Fibonacci numbers should be
linked to the ones above, yes, but also to the Tribonacci
seq, the Tetra- and Pentabonacci seqs, etc. This might
be difficult to put in the "definition" line.
2) there are already "transverse" classifications of the
seqs in the OEIS.

I agree -- but would like to insist: we don't "discover"
reality, we build it. Let's build more "artificial laws",
more combinatorics castles -- and if this helps someone
somewhere "in real life", so be it.

Best,
É.