[seqfan] Re: Knights-move Ulam-Warburton - nice problem
Paul Curtz
bpcrtz at free.fr
Mon Dec 17 12:34:13 CET 2018
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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