[seqfan] Re: Sequences taken from straight lines through the prime spiral.

Brad Klee bradklee at gmail.com
Sun Jul 29 20:58:14 CEST 2018

The ASCII drawings look nice, but the spiral plane figure does not seem
as special as Pascal's triangle (where linear intersections lead into
hypergeometric theory).  A135757 gives another example of Neil's
suggested construction besides A317187. Relative to core sequence
A000984, what is the novelty of A135757 ? Perhaps polynomial
indexing will lead to something interesting, but we should be careful
to avoid cranking out essentially duplicate entries.



On Fri, Jul 27, 2018 at 5:19 PM, Neil Sloane <njasloane at gmail.com> wrote:

> After reading David Sycamore's interesting message, I created A317186 to
> provide an overview of the sequences based on the two versions of the
> square spiral. (See the drawings of the two spirals in A317186.)
> There are many, certainly 24, natural sequences that one sees right away,
> all but one of which were already in the OEIS - the missing one (now
> A317186) gave me a peg to hang all the others on - as well as a general
> discussion.
>  I then created A317187 to show - following David S.'s message - an example
> of how these 24 sequences can be used to extract 24 subsequences from any
> existing sequence P by writing P in a square spiral. The particular example
> A317187
> is obtained by writing the primes in a spiral, and looking at those primes
> on the positive and negative X axis. In general this will extract
> subsequences from P that are indexed by quadratics.
> We could take P to be any of the 175 core sequences and we would get 4200
> or more potentially new sequences. I'm certainly not saying we should
> include all of them, but David has certainly (implicitly) pointed out a lot
> of sources for new sequences!
> Best regards
> Neil
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> On Thu, Jul 26, 2018 at 10:49 PM, Kevin Ryde via SeqFan <
> seqfan at list.seqfan.eu> wrote:
> > David Sycamore <djsycamore at yahoo.co.uk> writes:
> > >
> > > central vertical (N~S) ... indices (A267682) seems to
> > > be there, (Rule 201 etc).  Its not clear (to me at least) what
> > > A267682 is all about and if it could have any connection with the
> > > prime spiral.
> >
> > A267682 is another quadratic, if(n%2==0, n^2 - (n-2)/2, n^2 - (n-1)/2),
> > or pick your favourite (-1)^n etc for the cases.  Its rule 201 is full
> > rows except a fixed size alternating bit in the middle so cumulative odd
> > integers which is squares, less a bit.  As you noticed the even/odd
> > cases are interleaved A054556 and A033951 spiral spokes N,S.  I don't
> > suppose there'd be a connection to the spiral beyond triangle row
> > lengths vs spiral cycle length.
> >
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> >
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