[seqfan] Re: A269526, an infinite Sudoku-type array

[Bob Selcoe]

Hi,

Unless I'm missing something, I think there's a straightforward proof that 
the columns, rows and diagonals are permutations.

Let the array T(n,k) in A269526 start T(0,0).

Let j be the smallest number not yet appearing in Column K, and let F be the 
first cell in K where j may appear.  j only must avoid elements from prior 
(i.e, to the left of K) columns.  Since no element can repeat in any 
columns, the maximum number of elements that j must avoid is 3k (i.e, the 
elements in the row and two diagonals to the left of where j is being 
evaluated).  Therefore j must appear no later than 3k+1 places after F, and 
columns are permutations.

Same logic applies to the rows and diagonals, though the maximum number of 
places after F is different: 2n+k+1 for rows and n+2k+1 for diagonals.  The 
constraints probably can be tightened, but it's not necessary for a proof.

If this proof is acceptable I would like to add it as a comment in A269526.

Cheers,
Bob S.

--------------------------------------------------
From: "Neil Sloane" <njasloane at gmail.com>
Sent: Wednesday, June 29, 2016 1:24 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] A269526, an infinite Sudoku-type array

> Dear Seq Fans,  The following is a pretty interesting recent sequence:
>
> Array read by anti-diagonals upwards in which each term is the least
> positive value satisfying the condition that no row, column, or diagonal
> contains a repeated term.
>
> The sequence is A269526.  I just added the first three rows and the main
> diagonal as A274315 ff. (They all need b-files.)
>
> The array begins:
>
> 1, 3, 2, 6, 4, 5, 10, 11, 13, 8, 14, 18, 7, 20, 19, ...
> 2, 4, 5, 1, 8, 3, 6, 12, 14, 16, 7, 15, 17, 9, 22, ...
> 3, 1, 6, 2, 9, 7, 5, 4, 15, 17, 12, 19, 18, 21, 8, ...
> 4, 2, 3, 5, 1, 8, 9, 7, 16, 6, 18, 17, 11, 10, 23, ...
> 5, 7, 1, 4, 2, 6, 3, 15, 9, 10, 13, 8, 20, 14, 12, ...
> ...
>
> It seems very likely that every row, columns and diagonal (meaning
> diagonals parallel to the main diagonal) is a perm of the natural numbers,
> but I didn't try to find a proof.
>
> The first col is just 1,2,3,4,... but the next few columns could also be
> added as new? entries.
>
> There are a lot of other related sequences, for example, in row n, where
> does 1 appear?
>
> It is unusual to see such a nice array which is unrelated to any other
> sequence in the OEIS!  (But I didn't try Superseeker).
>
> This looks like a lovely problem crying out to be analyzed.
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
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