[seqfan] Re: Knight's tour on infinite board
yae9911 at gmail.com
Sat May 4 20:49:04 CEST 2019
For the name "Ulam spiral" see Neil's comment, e.g. in
https://oeis.org/A274923. Ulam spiral should only be used if the position
of the primes matters, otherwise we should use the unbiased term "square
spiral". For my taste, the most natural choice of a sequence pair
describing a counterclockwise square spiral would be
https://oeis.org/A274923 for y and https://oeis.org/A174344 for x, but
historically almost all variants can be found somewhere in our database.
Combining all spiral variants with the enumeration of various paths on the
2-d lattice or space filling curves like the knight's tour would be a can
of worms. I would rather not open this box of Pandora.
On Sat, May 4, 2019 at 8:17 PM Hans Havermann <gladhobo at bell.net> wrote:
> > It is dependant of course on the Ellermann spiral numbering scheme.
> Ellermann's "Ulam spiral" variant goes north, then spirals clockwise.
> Every Ulam spiral I've looked at in a search just now goes east, then
> spirals counter-clockwise. The knight's tour sequences would of course be
> different superimposed on a "proper" Ulam spiral.
> > The tour starts by filling in a central 5-by-5 square, starting in the
> square's center and ending in its SE corner. It then jumps 2 south and 1
> east and proceeds to circle the 5-by-5 square clockwise four times filling
> in the now-9-by-9 square and ending in its SE corner. And so on.
> If we flip the two-dimensional knight's tour over (in three-space) the
> spiral tour now runs counter-clockwise. Superimposed on Ellermann's Ulam
> spiral variant would generate new sequences differing from the eight that
> we have. So there is nothing really special about these eight sequences.
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