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

Author

Peter Kagey, Dec 27 2016

Keywords

• nonn
• tabl
• look

Comments

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)

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A003987, A269526, A274528.

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

Programs

Maple

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

Formula

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