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%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
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