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%I #13 Feb 25 2024 10:13:28
%S 1,0,0,4,0,0,72,0,0,108,0,0,4,0,0,180,0,3,216,0,0,252,0,0,264,0,0,0,0,
%T 0,360,0,5,396,0,0,432,0,0,468,0,0,0,0,0,868,0,5,576,0
%N Number of Dabbaghian-Wu pandiagonal Latin squares of order 2n+1 with the first row in order.
%C A pandiagonal Latin square is a Latin square in which the diagonal, antidiagonal and all broken diagonals and antidiagonals are transversals.
%C A Dabbaghian-Wu pandiagonal Latin square (see A368027) is a special type of pandiagonal Latin square (see A342306). Such squares are constructed from cyclic diagonal Latin squares (see A338562) for prime orders n=6k+1 (see Dabbaghian and Wu article) using a polynomial algorithm based on permutation of some values in Latin square. For other orders (25, 35, 49, ...) this algorithm also ensures correct pandiagonal Latin squares.
%H Vahid Dabbaghian and Tiankuang Wu, <a href="https://doi.org/10.1016/j.jda.2014.12.001">Constructing non-cyclic pandiagonal Latin squares of prime orders</a>, Journal of Discrete Algorithms, Vol. 30, 2015, pp. 70-77, doi: 10.1016/j.jda.2014.12.001.
%H <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e n=13=6*2+1 (prime order):
%e .
%e 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 2 3 0 1 11 12 8 4 10 7 5 6 9
%e 4 10 11 2 8 1 3 0 12 6 9 7 5
%e 11 5 9 7 10 0 12 1 3 2 8 4 6
%e 8 7 10 5 9 6 11 2 0 4 3 12 1
%e 12 0 4 6 7 2 9 10 5 11 1 8 3
%e 1 6 12 8 3 4 5 11 9 10 7 2 0
%e 9 2 3 4 12 8 1 6 7 5 0 10 11
%e 10 11 5 0 1 3 7 8 4 12 6 9 2
%e 5 9 1 11 2 10 0 12 6 8 4 3 7
%e 6 8 7 10 0 11 2 9 1 3 12 5 4
%e 7 4 6 12 5 9 10 3 2 0 11 1 8
%e 3 12 8 9 6 7 4 5 11 1 2 0 10
%e .
%e n=19=6*3+1 (prime order):
%e .
%e 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 2 3 0 1 11 12 8 4 10 7 5 6 9
%e 4 10 11 2 8 1 3 0 12 6 9 7 5
%e 11 5 9 7 10 0 12 1 3 2 8 4 6
%e 8 7 10 5 9 6 11 2 0 4 3 12 1
%e 12 0 4 6 7 2 9 10 5 11 1 8 3
%e 1 6 12 8 3 4 5 11 9 10 7 2 0
%e 9 2 3 4 12 8 1 6 7 5 0 10 11
%e 10 11 5 0 1 3 7 8 4 12 6 9 2
%e 5 9 1 11 2 10 0 12 6 8 4 3 7
%e 6 8 7 10 0 11 2 9 1 3 12 5 4
%e 7 4 6 12 5 9 10 3 2 0 11 1 8
%e 3 12 8 9 6 7 4 5 11 1 2 0 10
%e .
%e n=25=6*4+1 (nonprime order):
%e .
%e 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
%e 3 4 15 6 7 8 9 5 11 12 13 14 0 16 17 18 19 10 21 22 23 24 20 1 2
%e 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 21 22 23 24 5 1 2 3 4 20
%e 9 5 11 12 13 14 10 16 17 18 19 20 21 22 23 24 0 1 2 3 4 15 6 7 8
%e 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11
%e 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 18 19 10 21 22 23 24 20 1 2 3 4 15 6 7 8 9 5 11 12 13 14 0 16 17
%e 21 22 23 24 5 1 2 3 4 20 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0
%e 24 0 1 2 3 4 15 6 7 8 9 5 11 12 13 14 10 16 17 18 19 20 21 22 23
%e 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1
%e 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4
%e 8 9 5 11 12 13 14 0 16 17 18 19 10 21 22 23 24 20 1 2 3 4 15 6 7
%e 11 12 13 14 15 16 17 18 19 0 21 22 23 24 5 1 2 3 4 20 6 7 8 9 10
%e 14 10 16 17 18 19 20 21 22 23 24 0 1 2 3 4 15 6 7 8 9 5 11 12 13
%e 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16
%e 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
%e 23 24 20 1 2 3 4 15 6 7 8 9 5 11 12 13 14 0 16 17 18 19 10 21 22
%e 1 2 3 4 20 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 21 22 23 24 5
%e 4 15 6 7 8 9 5 11 12 13 14 10 16 17 18 19 20 21 22 23 24 0 1 2 3
%e 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6
%e 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9
%e 13 14 0 16 17 18 19 10 21 22 23 24 20 1 2 3 4 15 6 7 8 9 5 11 12
%e 16 17 18 19 0 21 22 23 24 5 1 2 3 4 20 6 7 8 9 10 11 12 13 14 15
%e 19 20 21 22 23 24 0 1 2 3 4 15 6 7 8 9 5 11 12 13 14 10 16 17 18
%e 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21
%Y Cf. A338562, A342306, A368027, A369380.
%K nonn,more
%O 1,4
%A _Eduard I. Vatutin_, Jan 22 2024
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