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

[Paul Curtz]

Hello all,
1,  1, 2, 4, 6, 11, 22, 43, 86, ... = a(n)
0,  1, 2, 2, 5, 11, 21, 43, 85, ...
1,  1, 0, 3, 6, 10, 22, 42, 86, ...
0, -1, 3, 3, 4, 12, 20, 44, 84, ...
-1, 4, 0, 1, 8,  8, 24, 40, 88, ...
5, -4, 1, 7, 0, 16, 16, 48, 80, ...
etc.
1) a(n) = A001045(n) + (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...). 
2) First upper diagonal: 1, 2, 3, 4, 8, 16, 32, 64, ... . Unknown. See A198633.
1,  2, 3, 4, 8, 16, 32, 64, ...
1,  1, 1, 4, 8, 16, 32, 64, ...
0,  0, 3, 4, 8, 16, 32, 64, ...
0,  3, 1, 4, 8, 16, 32, 64, ...
3, -2, 3, 4, 8, 16, 32, 64, ...
etc.
3) First upper diagonal: 2, 1, 4, 8, 16, 32, ... . Unknown. See 1, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... = A181440(n) -1. Unknown.

Paul



----- Mail original -----
De: "Hugo Pfoertner" <yae9911 at gmail.com>
À: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Envoyé: Lundi 17 Décembre 2018 00:30:57
Objet: [seqfan] Re: Knights-move Ulam-Warburton - nice problem

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/
>

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