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%I #48 Feb 20 2024 10:34:08
%S 1,0,0,24,0,0,72,0,0,108,0,0,4,0,0,180,0,3,216,0,0,252,0,0,264,0,0,0,
%T 0,0,360,0,5,396,0,0,432,0,0,468,0,0,0,0,0,868,0,5,576,0
%N Number of Dabbaghian-Wu pandiagonal Latin squares of order 2n+1.
%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 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 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2525">About the Dabbaghian-Wu pandiagonal Latin squares for non-prime orders</a> (in Russian).
%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 4 0 2 3 1 5 6 7 11 9 10 12 8
%e 11 12 1 10 6 2 4 5 3 7 8 9 0
%e 9 10 11 2 0 1 3 12 8 4 6 7 5
%e 6 8 9 7 11 12 0 4 2 3 5 1 10
%e 5 7 3 12 8 10 11 9 0 1 2 6 4
%e 3 4 8 6 7 9 5 1 10 12 0 11 2
%e 1 2 0 4 5 6 10 8 9 11 7 3 12
%e 0 9 5 1 3 4 2 6 7 8 12 10 11
%e 10 1 12 0 2 11 7 3 5 6 4 8 9
%e 8 6 10 11 12 3 1 2 4 0 9 5 7
%e 2 11 7 9 10 8 12 0 1 5 3 4 6
%e 7 5 6 8 4 0 9 11 12 10 1 2 3
%e 12 3 4 5 9 7 8 10 6 2 11 0 1
%e .
%e n=19=6*3+1 (prime order):
%e .
%e 8 0 2 3 4 6 17 7 1 9 10 11 12 13 14 15 16 5 18
%e 5 6 7 8 16 10 0 11 13 14 15 17 9 18 12 1 2 3 4
%e 10 4 12 13 14 15 16 17 18 0 8 2 11 3 5 6 7 9 1
%e 14 16 17 18 1 12 2 15 4 5 6 7 8 9 10 11 0 13 3
%e 1 2 3 11 5 14 6 8 9 10 12 4 13 7 15 16 17 18 0
%e 18 7 8 9 10 11 12 13 14 3 16 6 17 0 1 2 4 15 5
%e 11 12 13 15 7 16 10 18 0 1 2 3 4 5 6 14 8 17 9
%e 16 17 6 0 9 1 3 4 5 7 18 8 2 10 11 12 13 14 15
%e 2 3 4 5 6 7 8 9 17 11 1 12 14 15 16 18 10 0 13
%e 7 8 10 2 11 5 13 14 15 16 17 18 0 1 9 3 12 4 6
%e 12 1 14 4 15 17 18 0 2 13 3 16 5 6 7 8 9 10 11
%e 17 18 0 1 2 3 4 12 6 15 7 9 10 11 13 5 14 8 16
%e 3 5 16 6 0 8 9 10 11 12 13 14 15 4 17 7 18 1 2
%e 15 9 18 10 12 13 14 16 8 17 11 0 1 2 3 4 5 6 7
%e 13 14 15 16 17 18 7 1 10 2 4 5 6 8 0 9 3 11 12
%e 0 11 1 14 3 4 5 6 7 8 9 10 18 12 2 13 15 16 17
%e 4 13 5 7 8 9 11 3 12 6 14 15 16 17 18 0 1 2 10
%e 9 10 11 12 13 2 15 5 16 18 0 1 3 14 4 17 6 7 8
%e 6 15 9 17 18 0 1 2 3 4 5 13 7 16 8 10 11 12 14
%e .
%e n=25=6*4+1 (nonprime order):
%e .
%e 5 1 2 3 4 15 6 7 8 9 0 11 12 13 14 20 16 17 18 19 10 21 22 23 24
%e 3 4 20 6 7 8 9 15 11 12 13 14 5 16 17 18 19 0 21 22 23 24 10 1 2
%e 6 7 8 9 0 11 12 13 14 20 16 17 18 19 5 21 22 23 24 15 1 2 3 4 10
%e 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8
%e 12 13 14 5 16 17 18 19 0 21 22 23 24 15 1 2 3 4 10 6 7 8 9 20 11
%e 20 16 17 18 19 10 21 22 23 24 5 1 2 3 4 15 6 7 8 9 0 11 12 13 14
%e 18 19 0 21 22 23 24 10 1 2 3 4 20 6 7 8 9 15 11 12 13 14 5 16 17
%e 21 22 23 24 15 1 2 3 4 10 6 7 8 9 0 11 12 13 14 20 16 17 18 19 5
%e 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23
%e 2 3 4 10 6 7 8 9 20 11 12 13 14 5 16 17 18 19 0 21 22 23 24 15 1
%e 15 6 7 8 9 0 11 12 13 14 20 16 17 18 19 10 21 22 23 24 5 1 2 3 4
%e 8 9 15 11 12 13 14 5 16 17 18 19 0 21 22 23 24 10 1 2 3 4 20 6 7
%e 11 12 13 14 20 16 17 18 19 5 21 22 23 24 15 1 2 3 4 10 6 7 8 9 0
%e 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13
%e 17 18 19 0 21 22 23 24 15 1 2 3 4 10 6 7 8 9 20 11 12 13 14 5 16
%e 10 21 22 23 24 5 1 2 3 4 15 6 7 8 9 0 11 12 13 14 20 16 17 18 19
%e 23 24 10 1 2 3 4 20 6 7 8 9 15 11 12 13 14 5 16 17 18 19 0 21 22
%e 1 2 3 4 10 6 7 8 9 0 11 12 13 14 20 16 17 18 19 5 21 22 23 24 15
%e 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3
%e 7 8 9 20 11 12 13 14 5 16 17 18 19 0 21 22 23 24 15 1 2 3 4 10 6
%e 0 11 12 13 14 20 16 17 18 19 10 21 22 23 24 5 1 2 3 4 15 6 7 8 9
%e 13 14 5 16 17 18 19 0 21 22 23 24 10 1 2 3 4 20 6 7 8 9 15 11 12
%e 16 17 18 19 5 21 22 23 24 15 1 2 3 4 10 6 7 8 9 0 11 12 13 14 20
%e 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18
%e 22 23 24 15 1 2 3 4 10 6 7 8 9 20 11 12 13 14 5 16 17 18 19 0 21
%Y Cf. A338562, A369379, A369380, A342306.
%K nonn,more
%O 1,4
%A _Eduard I. Vatutin_, Dec 16 2023