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%I #13 May 26 2021 21:41:49
%S 1,0,240,20160,0,319334400,77127879628800,0
%N Number of 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 For orders 5, 7 and 11 all pandiagonal Latin squares are cyclic, so a(n) = A338562(n) for n < 6. For n=6 (order 13) this is not true (from Dabbaghian and Wu).
%C Pandiagonal Latin squares exist only for odd orders not divisible by 3. - _Andrew Howroyd_, May 26 2021
%H A.O.L. Atkin, L. Hay, and R. G. Larson, <a href="https://doi.org/10.1016/0898-1221(83)90130-X">Enumeration and construction of pandiagonal Latin squares of prime order</a>, Computers & Mathematics with Applications, Volume. 9, Iss. 2, 1983, pp. 267-292.
%H Vahid Dabbaghian and Tiankuang Wu, <a href="http://dx.doi.org/10.1016/j.jda.2014.12.001">Constructing non-cyclic pandiagonal Latin squares of prime orders</a>, Journal of Discrete Algorithms 30, 2015.
%F a(n) = A338620(n) * (2*n+1)!.
%e Example of a cyclic pandiagonal Latin square of order 5:
%e 0 1 2 3 4
%e 2 3 4 0 1
%e 4 0 1 2 3
%e 1 2 3 4 0
%e 3 4 0 1 2
%e Example of a cyclic pandiagonal Latin square of order 7:
%e 0 1 2 3 4 5 6
%e 2 3 4 5 6 0 1
%e 4 5 6 0 1 2 3
%e 6 0 1 2 3 4 5
%e 1 2 3 4 5 6 0
%e 3 4 5 6 0 1 2
%e 5 6 0 1 2 3 4
%e Example of a cyclic pandiagonal Latin square of order 11:
%e 0 1 2 3 4 5 6 7 8 9 10
%e 2 3 4 5 6 7 8 9 10 0 1
%e 4 5 6 7 8 9 10 0 1 2 3
%e 6 7 8 9 10 0 1 2 3 4 5
%e 8 9 10 0 1 2 3 4 5 6 7
%e 10 0 1 2 3 4 5 6 7 8 9
%e 1 2 3 4 5 6 7 8 9 10 0
%e 3 4 5 6 7 8 9 10 0 1 2
%e 5 6 7 8 9 10 0 1 2 3 4
%e 7 8 9 10 0 1 2 3 4 5 6
%e 9 10 0 1 2 3 4 5 6 7 8
%e For order 13 there is a square
%e 7 1 0 3 6 5 12 2 8 9 10 11 4
%e 2 3 4 10 0 7 6 9 12 11 5 8 1
%e 4 11 1 7 8 9 10 3 6 0 12 2 5
%e 6 5 8 11 10 4 7 0 1 2 3 9 12
%e 8 9 2 5 12 11 1 4 3 10 0 6 7
%e 3 6 12 0 1 2 8 11 5 4 7 10 9
%e 10 0 3 2 9 12 5 6 7 8 1 4 11
%e 1 7 10 4 3 6 9 8 2 5 11 12 0
%e 11 4 5 6 7 0 3 10 9 12 2 1 8
%e 5 8 7 1 4 10 11 12 0 6 9 3 2
%e 12 2 9 8 11 1 0 7 10 3 4 5 6
%e 9 10 11 12 5 8 2 1 4 7 6 0 3
%e 0 12 6 9 2 3 4 5 11 1 8 7 10
%e that is pandiagonal but not cyclic (Dabbaghian and Wu).
%Y Cf. A338562, A338620.
%K nonn,more,hard
%O 0,3
%A _Eduard I. Vatutin_, Mar 08 2021
%E Zero terms for even orders removed by _Andrew Howroyd_, May 26 2021