cp's OEIS Frontend

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

%I A280172 #99 Jun 11 2025 17:21:12
%S A280172 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,
%T A280172 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,
%U A280172 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
%N A280172 Lexicographically earliest table of positive integers read by antidiagonals such that no row or column contains a repeated term.
%C A280172 The table is symmetrical about the main diagonal.
%C A280172 The first row/column is A000027.
%C A280172 The second row/column is A103889.
%C A280172 The third row/column is A256008.
%C A280172 The fourth row/column is A113778.
%C A280172 Conjecture: The (2^k)-th antidiagonal consists entirely of 2^k.
%C A280172 Similar in spirit to A269526, A274528. - _N. J. A. Sloane_, Dec 27 2016
%C A280172 From _Daniel Forgues_, Sep 14 2019: (Start)
%C A280172 Plot of a(n) looks like a transform of a Sierpinski equilateral triangle.
%C A280172 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.
%C A280172 Conjecture: T(2n, k) = 2 * T(n, ceiling(k/2)), n >= 1, 1 <= k <= 2n. E.g.
%C A280172   row  5:      5,     3,    1,    3,     5
%C A280172   row 10:   10, 10, 6, 6, 2, 2, 6, 6, 10, 10 (End)
%C A280172 From _Daniel Forgues_, Sep 15 2019: (Start)
%C A280172 Conjectured algorithm for equilateral triangle (1-indexed rows and row terms), whose concatenated rows give this sequence: T(1, 1) = 1;
%C A280172 For each k >= 0, the height of the Sierpinski triangle is doubled:
%C A280172 * Left and right triangles: for 1 <= i <= 2^k, 1 <= j <= i:
%C A280172   T(2^k + i, j) = T(2^k + i, 2^k + i + 1 - j) = T(i, j) + 2^k;
%C A280172 * Central triangle: for 1 <= i <= 2^k - 1, 1 <= j <= i:
%C A280172   T(2^(k+1) - i, 2^k - i + j) = T(i, j).
%C A280172 Left and right triangles copies rows 1 to 2^k, terms augmented by 2^k.
%C A280172 Central triangle is mirrored through row 2^k.
%C A280172 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)
%H A280172 Peter Kagey, <a href="/A280172/b280172.txt">Table of n, a(n) for n = 1..32896</a> (first 256 rows, flattened)
%H A280172 Eric Gottlieb, Matjaž Krnc, and Peter Muršič, <a href="https://arxiv.org/abs/2506.04991">Nim on Integer Partitions and Hyperrectangles</a>, arXiv:2506.04991 [math.CO], 2025. See p. 16.
%H A280172 Peter Kagey, <a href="/A280172/a280172_1.png">Bitmap of first 2^10 = 1024 rows and columns</a>. (Black pixels correspond to numbers divisible by 3; white pixels to all other numbers.)
%H A280172 Rémy Sigrist, <a href="/A280172/a280172.png">Scatterplot of (n, a(n)*(a(n)+1)/2) for n = 1..2100225</a>
%F A280172 T(n, k) = ( (n-1) XOR (k-1) ) + 1 = A003987(n-1, k-1) + 1. - _Rémy Sigrist_, Sep 18 2019
%F A280172 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
%e A280172 As table (upper anti-triangular matrix) (concat. antidiagonals):
%e A280172   1 2 3 4 5 6 7 8
%e A280172   2 1 4 3 6 5 8
%e A280172   3 4 1 2 7 8
%e A280172   4 3 2 1 8
%e A280172   5 6 7 8
%e A280172   6 5 8
%e A280172   7 8
%e A280172   8
%e A280172 As equilateral triangle (concat. rows): (see formula section)
%e A280172          1
%e A280172         2 2
%e A280172        3 1 3
%e A280172       4 4 4 4
%e A280172      5 3 1 3 5
%e A280172     6 6 2 2 6 6
%e A280172    7 5 7 1 7 5 7
%e A280172   8 8 8 8 8 8 8 8
%e A280172 Lexicographically earliest equilateral triangle of positive integers read by rows such that no diagonal or antidiagonal contains a repeated term.
%p A280172 A280172 := (n, k) -> 1 + Bits:-Xor(k-1, n-k):
%p A280172 seq(print(seq(A280172(n, k), k=1..n)), n=1..14); # _Peter Luschny_, Sep 21 2019
%Y A280172 Cf. A003987, A269526, A274528.
%Y A280172 Rows (or columns) 1 to 4: A000027, A103889, A256008, A113778.
%K A280172 nonn,tabl,look
%O A280172 1,2
%A A280172 _Peter Kagey_, Dec 27 2016