[seqfan] Re: Strong Law of Small Numbers and the "Sequence in Context".

[Antti Karttunen]

jean-paul allouche jean-paul.allouche at imj-prg.fr wrote in:
http://list.seqfan.eu/pipermail/seqfan/2016-October/016849.html

>Hi; is this link useful then?
> http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf

>best
> jean-paul

Thanks a lot!

Now, that was published in 1988, when computers had just umpteenth of
the CPU power they have now. Nowadays it indeed would be very foolish
to propose any conjectures
(Guy: "Superficial similarities spawn spurious statements" and
"Capricious coincidences cause careless conjectures") based on some
pen-and-paper calculations of a few initial terms.

Almost any sequence (apart from some exceptions of course) can be
calculated so far that term-by-term matching with some existing
sequence is much less likely to be just a coincidence (well, because
then the numbers are no more "small").


Best regards,

Antti



Le 15/10/16 à 10:49, Antti Karttunen a écrit :
> Regarding Guy's Strong Law of Small Numbers
> (see e.g. https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers
> for a brief summary. I also had the paper printed/photocopied once,
> but I cannot find them now to check how much Wikipedia abridges or
> distorts its message)
>





On Sat, Oct 15, 2016 at 11:49 AM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> Regarding Guy's Strong Law of Small Numbers
> (see e.g. https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers
> for a brief summary. I also had the paper printed/photocopied once,
> but I cannot find them now to check how much Wikipedia abridges or
> distorts its message)
> I have come to realize that the law shouldn't be interpreted in such a
> way that there just exists many totally random pairs of sequences that
> have identical prefixes up to some value of n, and anybody who is
> inspired by such coincidences is just a fool.
>
> No, instead, I have observed that the strong law of small numbers is
> greatly helped if the two sequences sharing a common prefix also share
> a common thematic context.
>
> For all of my submissions, as a part of routine "post-submission
> checks", I always check the left and the right neighbour in the
> "Sequence in context" -line at Crossrefs-section. This often makes me
> aware of nice sequences (in general, interesting sequences have a mild
> tendency to cluster together), but sometimes there also is, or seems
> to be, some thematic connection with the sequence I submitted or
> edited.
>
> Of course this is clear in cases where the sequence A is counting some
> combinatorial structures say, and its neighbour B is counting some
> subset of those same combinatorial structures, but the difference does
> not show until after a while.
>
> Similarly if the sequence A is defined to list all numbers matching to
> some criteria X, and its neighbour B has a similar criteria, but with
> a bit more lax or restricted condition. (Maybe in future OEIS the
> server software would indicate with a special way those neighbours on
> the Context-line that _seem_ to be either supersets or subsets of the
> sequence? Maybe also using the data in b-files to make that informed
> guess.)
>
>
> But there are other cases also, where there probably is some kind of a
> connection (at least "statistical" if not anything else), like here:
>
> https://oeis.org/search?q=id%3AA277025%7Cid%3AA063952&sort=&language=&go=Search
>
> (Both sequences relate to expressing n as a sum of at most four squares).
>
>
> Best regards,
>
> Antti