Spiral sequences through the integer plane

[Joseph Biberstine]

I believe it stands to answer whether (some of) the coordinate sequences
are (in some transformation) in OEIS; are they?

-JRB

Jonathan Post wrote:
> 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
> <mailto: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
> 
>