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

[Bob Selcoe]

Hi Neil & Seqfans,

Forgive me in advance if my use of terminology and notation is substandard, 
but hopefully the ideas are clear enough.

Column 2 (i.e., terms a((j^2+j+4)/2), j>=1) is a permutation.   After 
a(3)=3, differences of successive terms follow the pattern a(n) = 3 [+1, -3, 
+1, +5], so a(5)=4, a(8)=1, a(12)=2, a(17)=7, a(23)=8, a(30)=5...

Similarly, Column 3 (i.e., terms a((j^2+j+6)/2), j>=2) appears to be a 
permutation, but with the pattern after a(6)=2 and a(9)=5 being 5 
[+1, -3, -2, +8, -5, +3, +1, +5, +1, -3, +1, -2, +8, -3, +1, +5].

I think that other similar cyclical difference patterns should hold for all 
Columns k (i.e., terms a(j^2+j+2k)/2), j>=k-1), all generating permutations, 
but I don't know how to formalize a proof.  Perhaps someone else can try, 
using this approach?

Also, note that differences for Column 1 are a 1-cycle ([+1]), Column 2 a 
4-cycle after the first term and Column 3 a 16-cycle after the second term. 
Perhaps cycle lengths are 4^(k-1) starting after j=k-1.  Could someone who 
knows how to program check this out?

Cheers,
Bob Selcoe


--------------------------------------------------
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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>