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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.

%I A288531 #46 Jun 14 2017 20:12:44
%S A288531 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,
%T A288531 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,
%U A288531 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
%N 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.
%C A288531 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.
%C A288531 Theorem 1: the middle diagonal gives A000012, the all 1's sequence.
%C A288531 Theorem 2: all 1's are in the middle diagonal.
%C A288531 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 A288531 Conjecture 3: every column is a permutation of the positive integers.
%C A288531 Conjecture 4: every diagonal is a permutation of the right border which gives the positive integers.
%H A288531 Alois P. Heinz, <a href="/A288531/b288531.txt">Rows n = 1..201, flattened</a>
%F A288531 T(n,k) = A288530(n-1,k-1) + 1.
%F A288531 T(n,n) = n.
%e A288531 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):
%e A288531 1,
%e A288531 3,   2,
%e A288531 4,   1,  3,
%e A288531 6,   5,  2,  4,
%e A288531 2,   3,  1,  6,  5,                      Every row is constructed
%e A288531 8,   7,  5,  2,  4,  6,              <---   from right to left.
%e A288531 10,  9,  4,  1,  3,  5,  7,
%e A288531 7,  11,  6,  5,  2, 10,  9,  8,
%e A288531 12, 10,  8,  3,  1,  4,  6,  7,  9,
%e A288531 5,   4, 13,  9,  7,  3, 12, 11,  8, 10,
%e A288531 16,  6, 15, 14, 13,  1,  8,  9,  7, 12, 11,
%e A288531 14, 13, 17, 16,  9,  7,  2,  6, 11,  8, 10, 12,
%e A288531 17, 15, 14,  8,  6, 12,  1,  4, 10,  7,  9, 11, 13,
%e A288531 ...
%e A288531 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!):
%e A288531 .
%e A288531 .                1,
%e A288531 .              3,  2,
%e A288531 .            4,  1,  3,
%e A288531 .          6,  5,  2,  4,
%e A288531 .        2,  3,  1,  6,  5,
%e A288531 .      8,  7,  5,  2,  4,  6,
%e A288531 .   10,  9,  4,  1,  3,  5,  7,
%e A288531 ...
%e A288531 Also the triangle may be reformatted for reading from left to right:
%e A288531 .
%e A288531 .                           1;
%e A288531 .                       2,  3;
%e A288531 .                   3,  1,  4;
%e A288531 .               4,  2,  5,  6;
%e A288531 .           5,  6,  1 , 3,  2;
%e A288531 .       6,  4,  2,  5,  7,  8;
%e A288531 .   7,  5,  3,  1,  4,  9, 10;
%e A288531 ...
%Y A288531 Middle diagonal gives A000012.
%Y A288531 Right border gives A000027.
%Y A288531 Indices of the 1's are in A001844.
%Y A288531 Cf. A288530 is the same triangle but with every entry minus 1.
%Y A288531 Other sequences of the same family are A269526, A274528, A274650, A274651, A274820, A274821, A286297.
%K A288531 nonn,tabl
%O A288531 1,2
%A A288531 _Omar E. Pol_, Jun 10 2017