A288530 - OEIS

%I #58 Sep 09 2017 16:11:26

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

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

%C Theorem 2: all zeros are in the middle diagonal.

%C 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.

%C Conjecture 3: every column is a permutation of the nonnegative integers.

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

%H Alois P. Heinz, <a href="/A288530/b288530.txt">Rows n = 0..200, flattened</a>

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

%F T(n,n) = n.

%e 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):

%e 0,

%e 2,   1,

%e 3,   0,  2,

%e 5,   4,  1,  3,

%e 1,   2,  0,  5,  4,                      Every row is constructed

%e 7,   6,  4,  1,  3,  5,              <---   from right to left.

%e 9,   8,  3,  0,  2,  4,  6,

%e 6,  10,  5,  4,  1,  9,  8,  7,

%e 11,  9,  7,  2,  0,  3,  5,  6,  8,

%e 4,   3, 12,  8,  6,  2, 11, 10,  7,  9,

%e 15,  5, 14, 13, 12,  0,  7,  8,  6, 11, 10,

%e 13, 12, 16, 15,  8,  6,  1,  5, 10,  7,  9, 11,

%e 16, 14, 13,  7,  5, 11,  0,  3,  9,  6,  8, 10, 12,

%e ...

%e 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!):

%e .

%e .              0,

%e .            2,  1,

%e ,          3,  0,  2,

%e .        5,  4,  1,  3,

%e .      1,  2,  0,  5,  4,

%e .    7,  6,  4,  1,  3,  5,

%e .  9,  8,  3,  0,  2,  4,  6,

%e ...

%e Also the triangle may be reformatted for reading from left to right:

%e .

%e .                           0;

%e .                       1,  2;

%e .                   2,  0,  3;

%e .               3,  1,  4,  5;

%e .           4,  5,  0 , 2,  1;

%e .       5,  3,  1,  4,  6,  7;

%e .   6,  4,  2,  0,  3,  8,  9;

%e ...

%Y Middle diagonal gives A000004.

%Y Right border gives A001477.

%Y Indices of the zeros are in A046092.

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

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

%K nonn,look,tabl

%O 0,3

%A _Omar E. Pol_, Jun 10 2017