A274650 -id:A274650 - cp's OEIS frontend

A288531 Triangle read by rows in reverse order: T(n,k), (1<=k<=n), in which each term is the least positive integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

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

Offset: 1

Omar E. Pol, Jun 10 2017

Note that the n-th row of this triangle is constructed from right to left, starting at the column n and ending at the column 1.
Theorem 1: the middle diagonal gives A000012, the all 1's sequence.
Theorem 2: all 1's are in the middle diagonal.
For the proofs of the theorems 1 and 2 see the proofs of the theorems 1 and 2 of A274650, because this is essentially the same problem.
Conjecture 3: every column is a permutation of the positive integers.
Conjecture 4: every diagonal is a permutation of the right border which gives the positive integers.

			Note that every row of the triangle is constructed from right to left, so the sequence is 1, 2, 3, 3, 1, 4,... (see below):
1,
3,   2,
4,   1,  3,
6,   5,  2,  4,
2,   3,  1,  6,  5,                      Every row is constructed
8,   7,  5,  2,  4,  6,              <---   from right to left.
10,  9,  4,  1,  3,  5,  7,
7,  11,  6,  5,  2, 10,  9,  8,
12, 10,  8,  3,  1,  4,  6,  7,  9,
5,   4, 13,  9,  7,  3, 12, 11,  8, 10,
16,  6, 15, 14, 13,  1,  8,  9,  7, 12, 11,
14, 13, 17, 16,  9,  7,  2,  6, 11,  8, 10, 12,
17, 15, 14,  8,  6, 12,  1,  4, 10,  7,  9, 11, 13,
...
The triangle may be reformatted as an isosceles triangle so that the all 1's sequence (A000012) appears in the central column (but note that this is NOT the way the triangle is constructed!):
.
.                1,
.              3,  2,
.            4,  1,  3,
.          6,  5,  2,  4,
.        2,  3,  1,  6,  5,
.      8,  7,  5,  2,  4,  6,
.   10,  9,  4,  1,  3,  5,  7,
...
Also the triangle may be reformatted for reading from left to right:
.
.                           1;
.                       2,  3;
.                   3,  1,  4;
.               4,  2,  5,  6;
.           5,  6,  1 , 3,  2;
.       6,  4,  2,  5,  7,  8;
.   7,  5,  3,  1,  4,  9, 10;
...

Alois P. Heinz, Rows n = 1..201, flattened

Middle diagonal gives A000012.
Right border gives A000027.
Indices of the 1's are in A001844.
Cf. A288530 is the same triangle but with every entry minus 1.
Other sequences of the same family are A269526, A274528, A274650, A274651, A274820, A274821, A286297.

T(n,k) = A288530(n-1,k-1) + 1.

T(n,n) = n.

A288424 Partial sums of A288384.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1

Offset: 0

Omar E. Pol, Jun 09 2017

nonn

It appears that the number of zeros is infinite.
Observation: for at least the first 110 terms the largest distance between two zeros that are between nonzero terms is 3.
Question: are there distances > 3?
From Hartmut F. W. Hoft, Jun 13 2017: (Start)
Yes: a(346..351) = (0,1,2,3,4,0).
Conjecture: a(n) >= 0 for all n >= 0, and a(n) is unbounded.
First occurrences: 3 = a(337) occurring 27 times; 4 = a(350) occurring 8 times; 5 = a(830) occurring 5 times; all through n=2500. (End)

Cf. A274650, A286294, A288384.

(* function a288384[] is defined in A288384 *)
a288424[n_] := Accumulate[a288384[n]]
a288424[104] (* data *) (* Hartmut F. W. Hoft, Jun 13 2017 *)

Signs reversed at the suggestion of Hartmut F. W. Hoft by Omar E. Pol, Jun 13 2017

A308178 Scan an infinite 45-degree triangular chessboard (cells (x,y) with 0 <= y <= x) by upwards antidiagonals, filling in each cell with the smallest nonnegative number already placed that cannot be seen by a chess queen at (x,y); sequence gives numbers along the successive antidiagonals.

Original entry on oeis.org

0, 1, 2, 3, 3, 0, 4, 1, 5, 5, 2, 4, 6, 7, 0, 2, 7, 4, 1, 3, 8, 5, 2, 4, 1, 9, 6, 3, 0, 2, 10, 11, 7, 1, 3, 6, 11, 8, 10, 9, 4, 5, 12, 9, 6, 8, 0, 10, 7, 13, 10, 12, 5, 6, 11, 8, 14, 15, 8, 7, 9, 3, 5, 11, 15, 12, 9, 6, 8, 0, 13, 10, 16, 13, 11, 12, 5, 1, 14, 7

Offset: 0

N. J. A. Sloane, May 28 2019

The 0's occur in positions (x,y) = (2k,k), k >= 0.
Column y=1 is A263313; the main diagonal is A308180.
After 13 steps, the y=2 column appears to become quasi-periodic with a saltus of 4. That is, the first differences appear to become periodic with period (-1, -2, 1, 6).
There is a very similar triangle in A274650.

			Start of chessboard showing antidiagonals 0 through 12:
y =  0, 1, 2, 3, 4, 5, 6, 7, ...
--------------------------------
x=0  0,
x=1  1, 3,
x=2  2, 0, 5,
x=3  3, 1, 4, 2,
x=4  4, 2, 0, 3, 1,
x=5  5, 7, 1, 4, 2, 6,
x=6  6, 4, 2, 0, 3, 5, 7,
x=7  7, 5, 3, 1, 4, 10, ...,
x=8  8, 6, 7, 9, 0, ...,
x=9  9, 11, 10, 8, ...,
x=10 10, 8, 6, ...,
x=11 11, 9, ...,
x=12 12, ...,
x=13 ...,
The first few antidiagonals are:
0,
1,
2, 3,
3, 0,
4, 1, 5,
5, 2, 4,
6, 7, 0, 2,
7, 4, 1, 3,
8, 5, 2, 4, 1,
9, 6, 3, 0, 2,
...

Rémy Sigrist, Table of n, a(n) for n = 0..10200 (antidiagonals for x+y = 0..200)
Rémy Sigrist, Colored representation of the first 1000 antidiagonals (black pixels correspond to zeros)
Rémy Sigrist, PARI program for A308178

Reading the triangle across rows gives A308179.

Cf. A263313, A274650, A308180

PARI
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See Links section.
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More terms from Rémy Sigrist, May 29 2019

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

A288531 Triangle read by rows in reverse order: T(n,k), (1<=k<=n), in which each term is the least positive integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

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A288424 Partial sums of A288384.

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A308178 Scan an infinite 45-degree triangular chessboard (cells (x,y) with 0 <= y <= x) by upwards antidiagonals, filling in each cell with the smallest nonnegative number already placed that cannot be seen by a chess queen at (x,y); sequence gives numbers along the successive antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula