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A280172 Lexicographically earliest table of positive integers read by antidiagonals such that no row or column contains a repeated term. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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)

LINKS

Peter Kagey, Table of n, a(n) for n = 1..32896 (first 256 rows, flattened)

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

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

EXAMPLE

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.

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

CROSSREFS

Cf. A003987, A269526, A274528.

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

Sequence in context: A191305 A227287 A289236 * A337942 A341458 A089913

Adjacent sequences:  A280169 A280170 A280171 * A280173 A280174 A280175

KEYWORD

nonn,tabl,look

AUTHOR

Peter Kagey, Dec 27 2016

STATUS

approved

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Last modified September 19 11:23 EDT 2021. Contains 347556 sequences. (Running on oeis4.)