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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.

%I A288530 #58 Sep 09 2017 16:11:26
%S A288530 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 A288530 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 A288530 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 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.
%C A288530 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 A288530 Theorem 1: the middle diagonal gives A000004, the all-zeros sequence.
%C A288530 Theorem 2: all zeros are in the middle diagonal.
%C A288530 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 A288530 Conjecture 3: every column is a permutation of the nonnegative integers.
%C A288530 Conjecture 4: every diagonal is a permutation of the right border which gives the nonnegative integers.
%H A288530 Alois P. Heinz, <a href="/A288530/b288530.txt">Rows n = 0..200, flattened</a>
%F A288530 T(n,k) = A288531(n+1, k+1) - 1.
%F A288530 T(n,n) = n.
%e A288530 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 A288530 0,
%e A288530 2,   1,
%e A288530 3,   0,  2,
%e A288530 5,   4,  1,  3,
%e A288530 1,   2,  0,  5,  4,                      Every row is constructed
%e A288530 7,   6,  4,  1,  3,  5,              <---   from right to left.
%e A288530 9,   8,  3,  0,  2,  4,  6,
%e A288530 6,  10,  5,  4,  1,  9,  8,  7,
%e A288530 11,  9,  7,  2,  0,  3,  5,  6,  8,
%e A288530 4,   3, 12,  8,  6,  2, 11, 10,  7,  9,
%e A288530 15,  5, 14, 13, 12,  0,  7,  8,  6, 11, 10,
%e A288530 13, 12, 16, 15,  8,  6,  1,  5, 10,  7,  9, 11,
%e A288530 16, 14, 13,  7,  5, 11,  0,  3,  9,  6,  8, 10, 12,
%e A288530 ...
%e A288530 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 A288530 .
%e A288530 .              0,
%e A288530 .            2,  1,
%e A288530 ,          3,  0,  2,
%e A288530 .        5,  4,  1,  3,
%e A288530 .      1,  2,  0,  5,  4,
%e A288530 .    7,  6,  4,  1,  3,  5,
%e A288530 .  9,  8,  3,  0,  2,  4,  6,
%e A288530 ...
%e A288530 Also the triangle may be reformatted for reading from left to right:
%e A288530 .
%e A288530 .                           0;
%e A288530 .                       1,  2;
%e A288530 .                   2,  0,  3;
%e A288530 .               3,  1,  4,  5;
%e A288530 .           4,  5,  0 , 2,  1;
%e A288530 .       5,  3,  1,  4,  6,  7;
%e A288530 .   6,  4,  2,  0,  3,  8,  9;
%e A288530 ...
%Y A288530 Middle diagonal gives A000004.
%Y A288530 Right border gives A001477.
%Y A288530 Indices of the zeros are in A046092.
%Y A288530 Cf. A288531 is the same triangle but with 1 added to every entry.
%Y A288530 Other sequences of the same family are A269526, A274528, A274650, A274651, A274820, A274821, A286297.
%K A288530 nonn,look,tabl
%O A288530 0,3
%A A288530 _Omar E. Pol_, Jun 10 2017