[seqfan] Re: Knights-move Ulam-Warburton - nice problem

Allan Wechsler acwacw at gmail.com
Mon Dec 17 02:58:57 CET 2018

```The graphs of the difference sequences (A319019, A322050) strongly suggest
some kind of self-similarity under the a map that takes 1+n to 1+2n. This
is a very vague hint as to a possible direction of work.

Would it be possible to change the scatterplot from logarithmic to linear?
I have a feeling the logarithmic representation is hiding a lot of
possibly-suggestive structure near the upper rim of the graph. Also, I see
a lot of vaguely log-n-shaped curves, which would be straight lines in a
linear representation.

On Sun, Dec 16, 2018 at 6:31 PM Hugo Pfoertner <yae9911 at gmail.com> wrote:

> Neil, all,
>
> using Remy's b-file I get the following for the number of agreeing terms in
> the rows:
> Start
>        agreeing
>                 first non-matching pair
>     3     1     5     1
>     5     1     7     5
>     9     2     6     3
>    17     4     8     5
>    33     6    17    15
>    65    11   145   141
>   129    22    73    69
>   257    43   734   726
>   513    86   349   341
>  1025   171  3579  3563
>  2049   342  1696  1680
>  4097   683 17810 17778
>  8193  1366  8394  8362
> 16385  2731 88553 88489
> 32769  5462 41665 41601
>
> After the row starting at position 33,
> the pattern of the match counts a(k)  seems to be
> 2*a(k-1) if a(k-1) is odd, else 2*a(k-1)-1
>
> Regards
> Hugo
>
> On Sun, Dec 16, 2018 at 6:08 PM Neil Sloane <njasloane at gmail.com> wrote:
>
> > Dear Seq Fans, Someone who watched the Terrific Toothpicks Numberphile
> > video on Youtube suggested a knights-move version.  I looked this up in
> the
> > OEIS of course and found that Remy Sigrist had already studied it. The
> > illustrations (excluding mine) are lovely, and so is the problem.
> > The original sequences are A319018, A319019; while trying to analyze it I
> > added three more last night, A322048, -049, -050.
> >
> > The main problem to crack is what is A322049.(which gives the growth from
> > the corners of the octagon after a power of 2 generations)?
> >
> > It would be nice to have a really big b-file (or a-file or both) for
> > A322050.
> >
> > Another question: look at the rows of the triangle in A322050. They are
> > converging to the sequence A322049. The number of terms in rows
> 1,2,3,4,...
> > that agree with the limit are 1,1,2,4,7,11,22 (I think). How does this
> > continue? This is not yet in the OEIS.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```