A269526 -id:A269526 - cp's OEIS frontend

A295567 Consider numbers k such that A295563(k) > k (see A295566); sequence lists the values A295563(k).

2, 5, 9, 10, 12, 13, 17, 19, 18, 23, 25, 22, 24, 28, 34, 30, 37, 39, 36, 38, 40, 42, 41, 46, 45, 44, 51, 50, 57, 52, 59, 49, 55, 61, 63, 62, 66, 65, 67, 72, 71, 58, 73, 76, 75, 69, 70, 86, 82, 83, 88, 87, 91, 90, 97, 92, 95, 99, 105, 104, 93, 102, 108, 98, 106, 112, 110, 111, 109, 116, 121, 117, 114

Offset: 1

N. J. A. Sloane, Nov 29 2017

nonn

Do the ratios A295567(k)/A295566(k) converge and if so what is the limit?

Rémy Sigrist, Table of n, a(n) for n = 1..50000

Cf. A269526, A274528, A295563, A295564, A295565, A295566,

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

Original entry on oeis.org

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

A274615 Third column (that is, the c=2 column) of array in A274528.

Original entry on oeis.org

1, 4, 5, 2, 0, 8, 3, 6, 7, 12, 13, 10, 11, 9, 17, 14, 15, 20, 21, 18, 16, 24, 19, 22, 23, 28, 29, 26, 27, 25, 33, 30, 31, 36, 37, 34, 32, 40, 35, 38, 39, 44, 45, 42, 43, 41, 49, 46, 47, 52, 53, 50, 48, 56, 51, 54, 55, 60, 61, 58, 59, 57, 65, 62, 63, 68, 69, 66

Offset: 0

N. J. A. Sloane, Jun 30 2016

nonn

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. A274528, A269526; equals A274614(n+1) - 1.

A[n_, k_] := A[n, k] = Module[{m, s}, If[n == 1 && k == 1, 0, s = Join[ Table[A[i, k], {i, 1, n - 1}], Table[A[n, j], {j, 1, k - 1}], Table[A[n - t, k - t], {t, 1, Min[n, k] - 1}], Table[A[n + j, k - j], {j, 1, k - 1}]]; For[m = 0, MemberQ[s, m], m++]; m]];
a[n_] := A[n + 1, 3];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 28 2020, after Alois P. Heinz in A269526 *)

This is a permutation of the nonnegative numbers (see the general proof in A269526).
It appears that the permutation is given by a(0)=0, and, for n >= 1, n = 16t+i (0 <= i <= 15) we have a(16t+i) = 16t + c_i, where [c_0, ..., c_15] = [-1,4,5,2,0,8,3,6,7,12,13,10,11,9,17,14]. - N. J. A. Sloane, Jul 01 2016, based on an email from Bob Selcoe, Jun 29 2016.
Equivalently, it appears that this sequence has g.f. = f/g where
f = 2*t^17 - 3*t^15 + 8*t^14 - 2*t^13 + t^12 - 3*t^11 + t^10 + 5*t^9 + t^8 + 3*t^7 - 5*t^6 + 8*t^5 - 2*t^4 - 3*t^3 + t^2 + 3*t + 1, and g = (1-t)*(1-t^16). - N. J. A. Sloane, Jul 06 2019

More terms from Alois P. Heinz, Jul 01 2016

A274632 First row of array in A274630.

Original entry on oeis.org

1, 3, 6, 2, 7, 5, 8, 4, 9, 10, 15, 13, 11, 18, 12, 20, 16, 22, 25, 27, 14, 28, 29, 31, 33, 34, 38, 32, 35, 36, 41, 40, 39, 17, 46, 43, 44, 52, 19, 48, 57, 50, 61, 53, 58, 21, 23, 24, 63, 65, 56, 60, 26, 71, 66, 68, 74, 78, 30, 77, 73, 80, 84, 69, 76, 82, 79, 89, 92, 85, 90, 87, 91, 93, 94, 37, 96

Offset: 1

N. J. A. Sloane, Jul 08 2016

nonn

Cf. A274630, A269526.

A275103 Hilbert curve constructed by greedy algorithm, such that each element is the smallest positive integer possible and that all rows, columns, and diagonals contain distinct numbers.

Original entry on oeis.org

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

Offset: 0

Kerry Mitchell, Jul 16 2016

nonn

The n-th cell has x-coordinates given by A059252 and y-coordinates given by A059253.

This idea is similar to A269526 and A274640, but for a different curve.

			The Hilbert curve begins:
  1,   4,   2,   3, ...
  2,   3,   5,   1, ...
  5,   6,   4,   2, ...
  4,   2,   1,   5, ...
...

Kerry Mitchell, Table of n, a(n) for n = 0..4095

Cf. A269526 uses antidiagonals instead of the Hilbert curve and A274640 uses a square spiral.

A274631 First column of array in A274630.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 7, 8, 10, 9, 11, 12, 13, 14, 16, 15, 17, 18, 19, 20, 22, 21, 24, 23, 26, 25, 28, 27, 30, 29, 32, 31, 34, 33, 36, 35, 38, 37, 39, 40, 42, 41, 44, 43, 46, 45, 48, 47, 50, 49, 52, 51, 54, 53, 56, 55, 58, 57, 59, 60, 62, 61, 64, 63, 66, 65, 68, 67, 70, 69, 72, 71, 74, 73, 76, 75, 78, 77, 79

Offset: 1

N. J. A. Sloane, Jul 08 2016

nonn

Cf. A269526, A274630.

First differs from A131042 at a(23). - Omar E. Pol, Jul 08 2016

A274633 Main diagonal of array in A274630.

Original entry on oeis.org

1, 5, 9, 6, 4, 14, 2, 19, 11, 3, 20, 15, 8, 10, 25, 22, 33, 27, 7, 28, 43, 29, 36, 17, 12, 13, 16, 62, 18, 40, 66, 35, 71, 59, 77, 74, 50, 45, 49, 85, 60, 21, 24, 58, 23, 99, 30, 78, 32, 31, 67, 103, 110, 37, 42, 34, 26, 38, 73, 83, 80, 39, 134, 47, 79, 105, 41, 95, 148, 106, 150, 102, 44, 92

Offset: 1

N. J. A. Sloane, Jul 08 2016

nonn

Cf. A274630, A269526.

A274652 Sum of n-th antidiagonal of the square array A274528.

Original entry on oeis.org

0, 3, 6, 12, 13, 25, 36, 43, 57, 67, 100, 134, 115, 163, 153, 186, 185, 248, 277, 258, 306, 366, 370, 381, 528, 471, 607, 662, 610, 706, 778, 760, 783, 782, 950, 970, 975, 1194, 1206, 1175, 1301, 1393, 1438, 1261, 1584, 1549, 1592, 1645, 1776, 1783, 2010, 1956, 1953, 2353, 2441, 2258, 2468, 2342, 2635, 2663

Offset: 0

Omar E. Pol, Jul 01 2016

nonn

Cf. A269526, A274528, A274529, A274530.

a(n) = A274530(n+1) - n - 1.

A334049 Triangular array read by rows in which each term is the least positive integer satisfying the condition that no row, diagonal, or antidiagonal contains a repeated term.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 1, 2, 6, 6, 7, 3, 8, 4, 5, 9, 8, 9, 5, 6, 1, 3, 7, 10, 11, 10, 11, 7, 12, 2, 8, 5, 6, 9, 13, 14, 12, 13, 9, 10, 14, 7, 1, 2, 15, 16, 11, 8, 17, 14, 15, 11, 16, 12, 9, 3, 4, 8, 10, 13, 17, 18, 7, 19, 16, 17, 13, 14, 10, 11, 8, 12, 1, 3, 15, 9

Offset: 1

Alec Jones and Peter Kagey, Sep 06 2020

An infinite Sudoku-type array.
Here, "diagonal" means a diagonal line with an inclination of -60 degrees, and "antidiagonal" means a diagonal line with an inclination of 60 degrees. Thus, the index-7 cell is in the same row as prior cells of indices 5 and 6, in the same diagonal as prior cells of indices 2 and 6, and in the same antidiagonal as the prior cell of index 4.
Triangular tiling is indexed as follows:
           *
          / \
         / 1 \
        *-----*
       / \ 3 / \
      / 2 \ / 4 \
     *-----*-----*
    / \ 6 / \ 8 / \
   / 5 \ / 7 \ / 9 \
  *-----*-----*-----*

			Triangle begins:
           *
          / \
         / 1 \
        *-----*
       / \ 3 / \
      / 2 \ / 4 \
     *-----*-----*
    / \ 5 / \ 2 / \
   / 4 \ / 1 \ / 6 \
  *-----*-----*-----*

Alec Jones, Table of n, a(n) for n = 1..5000

Triangular tiling analog of A269526.

Cf. A274821.

A335490 Isosceles triangle read by rows in which each term is the least positive integer satisfying the condition that no row, diagonal, or antidiagonal contains a repeated term.

Original entry on oeis.org

1, 2, 3, 3, 1, 2, 4, 2, 3, 5, 5, 6, 1, 4, 7, 6, 4, 5, 7, 8, 9, 7, 5, 6, 1, 4, 10, 8, 8, 9, 4, 2, 3, 5, 6, 10, 9, 7, 8, 3, 1, 2, 10, 5, 4, 10, 8, 9, 6, 2, 3, 7, 11, 12, 13, 11, 12, 7, 10, 5, 1, 9, 8, 6, 14, 15, 12, 10, 11, 13, 6, 4, 14, 7, 9, 8, 16, 17, 13, 11

Offset: 1

Alec Jones and Peter Kagey, Sep 12 2020

The n-th instance of 1 occurs at index A001844(n-1).

Records occur at 1, 2, 3, 7, 10, 12, 15, 20, 21, 27, 53, 54, 55, 65, ...

			Triangle begins:
       1
      2 3
     3 1 2
    4 2 3 5
   5 6 1 4 7
  6 4 X ...
The value for X is 5 because 1, 2, and 3 are on the diagonal; 4 and 6 are on the antidiagonal; and 4 and 6 are in the row. Therefore 5 is the smallest value that can be inserted so that no diagonal, antidiagonal, or row contains a repeated term.

Analogs for other tilings: A269526 (square), A334049 (triangular).

Cf. A001844, A274650, A274651, A288530, A288531, A296339.

a(n) = A296339(n-1) + 1. - Rémy Sigrist, Sep 13 2020

cp's OEIS Frontend

A295567 Consider numbers k such that A295563(k) > k (see A295566); sequence lists the values A295563(k).

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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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A274615 Third column (that is, the c=2 column) of array in A274528.

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A274632 First row of array in A274630.

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A275103 Hilbert curve constructed by greedy algorithm, such that each element is the smallest positive integer possible and that all rows, columns, and diagonals contain distinct numbers.

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A274631 First column of array in A274630.

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A274633 Main diagonal of array in A274630.

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A274652 Sum of n-th antidiagonal of the square array A274528.

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A334049 Triangular array read by rows in which each term is the least positive integer satisfying the condition that no row, diagonal, or antidiagonal contains a repeated term.

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A335490 Isosceles triangle read by rows in which each term is the least positive integer satisfying the condition that no row, diagonal, or antidiagonal contains a repeated term.

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