Spiral sequences through the integer plane

[Jonathan Post]

Spiral centered on "1" rather than (0,0) -- the two are isomorphic. Nothing
more than quadratic equations.

Search OEIS for the phrase "ulam spiral."

 A004526 <http://www.research.att.com/%7Enjas/sequences/A004526>
 Integers repeated.
 A033638 <http://www.research.att.com/%7Enjas/sequences/A033638>
 Quarter-squares plus 1 (i.e.
A002620<http://www.research.att.com/%7Enjas/sequences/A002620>+ 1).


  A073577 <http://www.research.att.com/%7Enjas/sequences/A073577>
 4n^2+4n-1.  A063826 <http://www.research.att.com/%7Enjas/sequences/A063826>
 Let 1, 2, 3, 4 represent moves to the right, down, left and up; this
sequence describes the movements in the Ulam Spiral.
A078765<http://www.research.att.com/%7Enjas/sequences/A078765>
 Prime numbers occurring at integer Pythagorean distance (radius) from 1 in
Ulam square prime-spiral. Primes on axes are excluded.
A078784<http://www.research.att.com/%7Enjas/sequences/A078784>
 Primes on axis of Ulam square prime-spiral (with rows ... 7 8 9 / 6 1 2 / 5
4 3 /... ) with origin at (1).
A115258<http://www.research.att.com/%7Enjas/sequences/A115258>
 Isolated primes in Ulam ' s lattice (1,2,... in spiral).
A113688<http://www.research.att.com/%7Enjas/sequences/A113688>
 Isolated semiprimes in the semiprime spiral.
A113689<http://www.research.att.com/%7Enjas/sequences/A113689>
 Number of semiprimes in clumps of size >1 through n^2 in the semiprime
spiral.
Plus those where the spiral is in the triangular lattice.

Search OEIS for "hexagonal spiral" -- as in
 A063178 <http://www.research.att.com/%7Enjas/sequences/A063178>
 Hexagonal spiral sequence: sequence is written as a hexagonal spiral around
a `dummy' center, each entry is the sum of the row in the previous direction
containing the previous entry.
A113519<http://www.research.att.com/%7Enjas/sequences/A113519>
 Semiprimes in first spoke of a hexagonal spiral
(A056105<http://www.research.att.com/%7Enjas/sequences/A056105>
).  A056106 <http://www.research.att.com/%7Enjas/sequences/A056106>
 Second spoke of a hexagonal spiral.
A056107<http://www.research.att.com/%7Enjas/sequences/A056107>
 Third spoke of a hexagonal spiral.
A056108<http://www.research.att.com/%7Enjas/sequences/A056108>
 Fourth spoke of a hexagonal spiral.
A056109<http://www.research.att.com/%7Enjas/sequences/A056109>
 Fifth spoke of a hexagonal spiral.
And so forth.  Lot's of hotlinks to spirals outside OEIS.  Well-known, and
often rediscovered.  Note, for instance, Sir Arthur C. Clarke having
virtually anticipated the Ulam spiral in "The City and the Stars."

On 10/1/06, Joseph Biberstine <jrbibers at indiana.edu> wrote:
>
> I'm shocked that these seem to be absent.  Consider the coordinates of a
> path spirally traversing the integer plane (principal quadrant or whole)
> any of the usual ways.  These all seem to be absent from EIS.  Am I right?
>
> For example, spanning the principal quadrant in this way in either of
> the two departures from the origin, for example:
> {{0,0},{0,1},{1,1},{1,0},{2,0},{2,1},{2,2},{1,2},{0,2},{0,3}, ...}
>
> Or spiraling in either handedness from any of four departures from the
> origin, for example:
> {{0,0},{1,0},{1,1},{0,1},{-1,1},{-1,0},{-1,-1},{0,-1},{1,-1},...}
>
> Are these in EIS and if not can you provide a closed formula for the
> n-th coordinate in any matter of traversal?
>
> -JRB
>
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