[seqfan] OEIS as a combinatorics castle

Sat Dec 12 18:12:11 CET 2015

On Sat, Dec 12, 2015 at 1:00 PM,  <seqfan-request at list.seqfan.eu> wrote:

>
> Message: 11
> Date: Thu, 10 Dec 2015 15:34:07 +0100
> From: Eric Angelini <Eric.Angelini at kntv.be>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Cc: "alexandre.wajnberg at skynet.be" <alexandre.wajnberg at skynet.be>
> Subject: [seqfan] OEIS as a combinatorics castle
> Message-ID:
>         <8B00BFBA136BAB43AD27F9EDC3758F03BD32D0E1C9 at KNTVSRV01.kntv.local>
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>
> 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.

Interesting musings and a point of view, Eric.
However, I think myself that the best "artificial laws" are those that
feel most "natural" to me.
Even though I like to rearrange the numbers, still for my taste, most
of the "greedy algorithm permutations" (like EKG-sequence or any other
of the type "for the next term choose the least number which satisfies
certain condition involving the previously constructed prefix of the
sequence") are _too_ self-referential, not that much involved with a
larger universe of sequences. Of course many of them offer very nice
problems (like e.g. whether they are permutations in the first place)
for people who like tough nuts, but as I have quite a limited
brain-power, I abstain from that sport.

Instead, I keep on finding isomorphisms and bijections between
"naturally" occurring entities (like e.g. partitions, etc, present in
prime factorizations, run lengths of base-2 representation, and so
on), and then create sequences by cross-referencing them. I guess this
is all trivial from the view point of real mathematicians, but I like
it. Also, staying in the "natural landscape", or deviating only a
little from it, leaves always a possibility of surprising serendipity
years later, when somebody else (or my future self) bumps to same
sequences in an apparently different context (different enough that it
doesn't feel trivial!). And I think that's the most important point of
doing this publicly, and not just collecting these to one's own hard
drive. Unexpected social connections with many interesting and
enlightened people.

But of course I like also to construct more "artificial" things, but
still keeping in mind the first principle, that "the chosen components
should fit well together".

For example, various number sieves is an interesting phenomenon, also
culturally (Lucky numbers from Ulam, and so on).
Here's an example how two of them (Sieve of Eratosthenes and Ludic
numbers sieve) co-create something visually nice:
https://oeis.org/A255422/graph (where does that periodicity of those
gentle waves ultimately come from?)

Then, just last night, after I had a nights before edited Peter
Kagey's new sequence involving Ulam-spiral (Ulam again...), and
created a few indexing sequences to be able to define it more exactly,
I suddenly realized that the other indexing sequence could be used to
create "Spironacci numbers". And indeed, it was already in OEIS with
almost the same name, "Spiro-Fibonacci numbers":
https://oeis.org/A078510
(submitted by Neil Fernandez, Jan 05 2003).

Now, this is an interesting case. For example, where would you place
it in your continuum of "natural" ... "artificial" laws?
Of course it's similar to Fibonacci-like recurrences in that the next
term is computed from two earlier terms of the same sequence (as their
sum), but here the distance to the lesser of these keeps on growing
continually.
Also, it seems like its growth-rate is revving up quite slowly,
compared to "constantly-relatively-indiced" recurrences.

Artificial or not, I think it might be nice starting material to be
used in a few basic templates I have employed in my other
"construction projects".

Best regards,

Antti

>
> Best,
> É.
>