[seqfan] Re: Knight's tour on infinite board
yae9911 at gmail.com
Sun May 5 23:08:33 CEST 2019
@Maximilian: One has to be careful with the path encoding, because the
direction changes against the spine of the previous path segment are not
multiples of 360/8 degrees, but are dependent on the orientation of the
previous move. I have used a similar encoding using letters in the
illustrations of a couple of knight's path sequences, e.g.
https://oeis.org/A323132 and the illustration html file linked there.
I have now understood the method of construction used in A068608-A068615.
To get rid of the spiral encoding, I've made 2 new sequences
https://oeis.org/draft/A306659 and https://oeis.org/draft/A306660 with a
description of the method in the comment of A306659. At the moment, I have
only a Fortran program, but I will try to covert it into a PARI code. In an
adapted version it should also work for the A068608 ff sequence family.
On Sun, May 5, 2019 at 5:54 AM M. F. Hasler <seqfan at hasler.fr> wrote:
> On Sat, May 4, 2019, 20:50 Hans Havermann <gladhobo at bell.net> wrote:
> HP: "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've actually thought about it a bit. There appear to be 8 square spiral
> variants: First step east, north, west, or south, followed by either a
> clockwise or counterclockwise numbering(.…)
> These tours could also be coded using "relative" co-ordinates rather than
> absolute (x,y) coordinates.
> A very natural encoding could be to denote by 0, ±1, ±2..., respectively, a
> move in the same direction as the preceding one, or involving a "turn" by
> that multiple of 360°/8 to the left or right, respectively.
> That way the encoding of the Knight tour becomes independent of the
> starting direction (and the encoding more compact). Rotational invariance
> is intrinsic and invariance wrt reflections amounts to changing the sign of
> all terms.
> - Maximilian
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