[seqfan] Strong Law of Small Numbers and the "Sequence in Context".
Antti Karttunen
antti.karttunen at gmail.com
Sat Oct 15 10:49:31 CEST 2016
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
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