A308880 Irregular array read by rows: row k (k>=1) contains k^2 numbers, formed by filling in a k X k square by rows so entries in all rows, columns, diagonals, antidiagonals are distinct, and then reading that square across rows.

0, 0, 1, 2, 3, 0, 1, 2, 2, 3, 0, 1, 4, 5, 0, 1, 2, 3, 2, 3, 0, 1, 1, 4, 5, 2, 5, 0, 1, 4, 0, 1, 2, 3, 4, 2, 3, 0, 1, 5, 1, 4, 5, 2, 0, 5, 0, 1, 4, 3, 3, 6, 7, 0, 1, 0, 1, 2, 3, 4, 5, 2, 3, 0, 1, 6, 7, 1, 4, 5, 2, 0, 8, 5, 0, 1, 4, 3, 6, 3, 7, 6, 0, 1, 4, 4, 2, 9, 5, 7, 10

Offset

1,4

Comments

The second row of the k X k square converges to A004443 as k increases.

When filling in the k X k square, always choose the smallest possible number. Each k X k square is uniquely determined.

Each k X k square read downwards by antidiagonals up to and including the main antidiagonal is A274528(1..k*(k+1)/2). - I. V. Serov, Jun 30 2019, following an argument by Bernard Schott.

Links

I. V. Serov, Rows of first 32 squares, flattened (There are 1^2+2^2+...+32^2 = 11440 entries.)

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Example

The first eight squares are (here A=10, B=11, C=12):
0
--------
01
23
--------
012
230
145
--------
0123
2301
1452
5014
--------
01234
23015
14520
50143
36701
--------
012345
230167
145208
501436
376014
42957A
--------
0123456
2301674
1452083
5014362
3780145
4265798
9548237
--------
01234567
23016745
14520836
50143628
37801459
42675983
9548237A
A836BC92
--------
Concatenating the rows of these squares gives the sequence.

Prog

(MATLAB)
A308880 = [];
A308881 = [];
for n = 1:oo;
M = [0:(n-1)
     zeros(n-1, n-0)*NaN];
for i = 2:n; for j = 1:n; M = Mnext(M, n, i, j); end; end
A308880 = [A308880 reshape(M', 1, n^2)];
A308881 = [A308881 reshape(M , 1, n^2)];
end
function [M] = Mnext(M, n, i, j);
row = M(i, 1:j-1);
col = M(1:i-1, j);
dim = diag(       M, j-i);
dia = diag(fliplr(M), n-i-j+1);
X = ([row col' dim' dia']);
for m = 0:length(X)-1; if isempty(find(X==m)); break; end; end;
M(i, j) = m;
end
% I. V. Serov, Jun 30 2019

Crossrefs

Cf. A004443, A308881, A274528.

Sequence in context: A308898 A106728 A292603 * A319047 A276335 A189480

Adjacent sequences: A308877 A308878 A308879 * A308881 A308882 A308883

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jun 29 2019

Status

approved