A274528 -id:A274528 - cp's OEIS frontend

A280172 Lexicographically earliest table of positive integers read by antidiagonals such that no row or column contains a repeated term.

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

Offset: 1

Peter Kagey, Dec 27 2016

The table is symmetrical about the main diagonal.
The first row/column is A000027.
The second row/column is A103889.
The third row/column is A256008.
The fourth row/column is A113778.
Conjecture: The (2^k)-th antidiagonal consists entirely of 2^k.
Similar in spirit to A269526, A274528. - N. J. A. Sloane, Dec 27 2016
From Daniel Forgues, Sep 14 2019: (Start)
Plot of a(n) looks like a transform of a Sierpinski equilateral triangle.
Considering t(a(n)) = a(n)*(a(n)+1)/2: top edge of plot would be linear, but left & right sides of [concave curved] triangles would grow/decrease quadratically. a(n), a univalued sequence, tries to plot a Sierpinski triangle, which requires a multivalued sequence: a(n) uses t(2^k) terms to draw a Sierpinski triangle of width & height 2^k.
Conjecture: T(2n, k) = 2 * T(n, ceiling(k/2)), n >= 1, 1 <= k <= 2n. E.g.
  row  5:      5,     3,    1,    3,     5
  row 10:   10, 10, 6, 6, 2, 2, 6, 6, 10, 10 (End)
From Daniel Forgues, Sep 15 2019: (Start)
Conjectured algorithm for equilateral triangle (1-indexed rows and row terms), whose concatenated rows give this sequence: T(1, 1) = 1;
For each k >= 0, the height of the Sierpinski triangle is doubled:
* Left and right triangles: for 1 <= i <= 2^k, 1 <= j <= i:
  T(2^k + i, j) = T(2^k + i, 2^k + i + 1 - j) = T(i, j) + 2^k;
* Central triangle: for 1 <= i <= 2^k - 1, 1 <= j <= i:
  T(2^(k+1) - i, 2^k - i + j) = T(i, j).
Left and right triangles copies rows 1 to 2^k, terms augmented by 2^k.
Central triangle is mirrored through row 2^k.
When n is t(2^k), k >= 0, i.e., a triangular number with index a power of 2, a phase of the Sierpinski triangle plot is neatly completed. (End)

			As table (upper anti-triangular matrix) (concat. antidiagonals):
  1 2 3 4 5 6 7 8
  2 1 4 3 6 5 8
  3 4 1 2 7 8
  4 3 2 1 8
  5 6 7 8
  6 5 8
  7 8
  8
As equilateral triangle (concat. rows): (see formula section)
         1
        2 2
       3 1 3
      4 4 4 4
     5 3 1 3 5
    6 6 2 2 6 6
   7 5 7 1 7 5 7
  8 8 8 8 8 8 8 8
Lexicographically earliest equilateral triangle of positive integers read by rows such that no diagonal or antidiagonal contains a repeated term.

Peter Kagey, Table of n, a(n) for n = 1..32896 (first 256 rows, flattened)
Eric Gottlieb, Matjaž Krnc, and Peter Muršič, Nim on Integer Partitions and Hyperrectangles, arXiv:2506.04991 [math.CO], 2025. See p. 16.
Peter Kagey, Bitmap of first 2^10 = 1024 rows and columns. (Black pixels correspond to numbers divisible by 3; white pixels to all other numbers.)
Rémy Sigrist, Scatterplot of (n, a(n)*(a(n)+1)/2) for n = 1..2100225

Cf. A003987, A269526, A274528.

Rows (or columns) 1 to 4: A000027, A103889, A256008, A113778.

A280172 := (n, k) -> 1 + Bits:-Xor(k-1, n-k):
seq(print(seq(A280172(n, k), k=1..n)), n=1..14); # Peter Luschny, Sep 21 2019

T(n, k) = ( (n-1) XOR (k-1) ) + 1 = A003987(n-1, k-1) + 1. - Rémy Sigrist, Sep 18 2019

a(n) = T(row, n - t(row - 1)), n >= 1, where row = ceiling((-1 + sqrt(1 + 8*n))/2) and t(i) = i*(i+1)/2. - Daniel Forgues, Sep 20 2019

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.

Original entry on oeis.org

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

N. J. A. Sloane, Jun 29 2019

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.

			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.

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.

Cf. A004443, A308881, A274528.

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

A273139 Where records occur in A269526.

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 17, 20, 26, 28, 36, 44, 45, 54, 64, 65, 76, 78, 103, 105, 134, 135, 168, 171, 189, 190, 230, 252, 253, 275, 298, 299, 323, 324, 325, 376, 377, 378, 405, 406, 463, 493, 494, 627, 628, 629, 630, 736, 737, 738, 740, 741, 779, 859, 899, 902, 944, 946, 1033, 1035, 1171, 1176, 1223, 1225

Offset: 1

Omar E. Pol, Jul 02 2016

nonn

Cf. A269526, A273138, A274528.

A317191 Fill an n X n square array T(j,k), 1<=j<=n, 1=k<=n, by antidiagonals upwards in which each term is the least nonnegative integer satisfying the condition that no row, column, diagonal, or antidiagonal contains a repeated term; a(n) = T(n,n).

Original entry on oeis.org

0, 3, 5, 4, 1, 10, 7, 2, 6, 8, 15, 12, 19, 17, 22, 23, 12, 26, 11, 31, 32, 12, 35, 10, 37, 42, 40, 45, 33, 49, 18, 17, 20, 53, 16, 51, 59, 18, 59, 60, 58, 64, 69, 69, 38, 29, 74, 26, 68, 78, 80, 36, 30, 33, 41, 39, 32, 33, 92, 41, 38, 89, 32, 35

Offset: 1

N. J. A. Sloane and Doron Zeilberger, Jul 30 2018

nonn

This is the analog for an n X n board of the sequence A317190 (which is the main diagonal when we fill in the whole of the fourth quadrant in this way).

			For n=3 the array T is
0 2 1
1 3 4
2 0 5
so a(3) = T(3,3) = 5.
For n=6 the array T is
0 2 1 5 3 4
1 3 4 0 7 2
2 0 5 1 6 9
3 1 2 4 0 5
4 6 0 3 1 7
5 7 8 6 4 10
so a(6) = T(6,6) = 10. This is the first time this sequence differs from A317190.

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.

Cf. A317190, A274318, A269529, A274528.

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A280172 Lexicographically earliest table of positive integers read by antidiagonals such that no row or column contains a repeated term.

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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.

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A273139 Where records occur in A269526.

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A317191 Fill an n X n square array T(j,k), 1<=j<=n, 1=k<=n, by antidiagonals upwards in which each term is the least nonnegative integer satisfying the condition that no row, column, diagonal, or antidiagonal contains a repeated term; a(n) = T(n,n).

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