

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
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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 (1indexed 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) = ( (n1) XOR (k1) ) + 1 = A003987(n1, k1) + 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 antitriangular 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(k1, nk):
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



