A288530 Triangle read by rows in reverse order: T(n,k), (0 <= k <= n), in which each term is the least nonnegative integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

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

Offset

0,3

Comments

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 0.

Theorem 1: the middle diagonal gives A000004, the all-zeros sequence.

Theorem 2: all zeros 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 nonnegative integers.

Conjecture 4: every diagonal is a permutation of the right border which gives the nonnegative integers.

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

T(n,k) = A288531(n+1, k+1) - 1.

T(n,n) = n.

Example

Note that every row of the triangle is constructed from right to left, so the sequence is 0, 1, 2, 2, 0, 3, ... (see below):
0,
2,   1,
3,   0,  2,
5,   4,  1,  3,
1,   2,  0,  5,  4,                      Every row is constructed
7,   6,  4,  1,  3,  5,              <---   from right to left.
9,   8,  3,  0,  2,  4,  6,
6,  10,  5,  4,  1,  9,  8,  7,
11,  9,  7,  2,  0,  3,  5,  6,  8,
4,   3, 12,  8,  6,  2, 11, 10,  7,  9,
15,  5, 14, 13, 12,  0,  7,  8,  6, 11, 10,
13, 12, 16, 15,  8,  6,  1,  5, 10,  7,  9, 11,
16, 14, 13,  7,  5, 11,  0,  3,  9,  6,  8, 10, 12,
...
The triangle may be reformatted as an isosceles triangle so that the all-zeros sequence (A000004) appears in the central column (but note that this is NOT the way the triangle is constructed!):
.
.              0,
.            2,  1,
,          3,  0,  2,
.        5,  4,  1,  3,
.      1,  2,  0,  5,  4,
.    7,  6,  4,  1,  3,  5,
.  9,  8,  3,  0,  2,  4,  6,
...
Also the triangle may be reformatted for reading from left to right:
.
.                           0;
.                       1,  2;
.                   2,  0,  3;
.               3,  1,  4,  5;
.           4,  5,  0 , 2,  1;
.       5,  3,  1,  4,  6,  7;
.   6,  4,  2,  0,  3,  8,  9;
...

Crossrefs

Middle diagonal gives A000004.

Right border gives A001477.

Indices of the zeros are in A046092.

Cf. A288531 is the same triangle but with 1 added to every entry.

Other sequences of the same family are A269526, A274528, A274650, A274651, A274820, A274821, A286297.

Sequence in context: A095731 A048142 A071426 * A295675 A135356 A259016

Adjacent sequences: A288527 A288528 A288529 * A288531 A288532 A288533

Keyword

nonn,look,tabl

Author

Omar E. Pol, Jun 10 2017

Status

approved