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

[Zak Seidov]

 BTW See my old spiral (old hat?):

 15, 14, 13, 11, 10, 12, 6, 8,
 4, 11, 9, 2, 8, 7, 10, 12,
 7, 3, 5, 6, 1, 2, 4, 9,
 8, 1, 2, 4, 3, 6, 5, 11,
 6, 5, 3, 1, 2, 4, 9, 7,
 3, 2, 4, 5, 6, 1, 8, 10,
 9, 6, 1, 3, 4, 5, 7, 2,
 5, 7, 8, 10, 9, 11, 12, 3

Each diagonal and each row/column consist of distinct numbers.  
Sequence of numbers along the counter-clokwise spiral: 
1, 2, 3, 4, 2,3,4,5,6,1,4,6,2,1,6,5,2,3,4.
See better picture in my post Old Spiral in 
https://www.facebook.com/zak.seidov  

Sorry if formatting/exposition  is bad:(




>Четверг, 30 июня 2016, 0:26 +03:00 от "Bob Selcoe" <rselcoe at entouchonline.net>:
>
>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
>>
>> --
>> Seqfan Mailing list -  http://list.seqfan.eu/
>> 
>
>--
>Seqfan Mailing list -  http://list.seqfan.eu/