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

[Neil Sloane]

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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