cp's OEIS Frontend

A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.

%I A370672 #12 Mar 14 2024 05:31:51
%S A370672 1,0,10,28,0,88,130,0,238,304,0,460,250,0,754,868,0,280,1258,0,1558,
%T A370672 1720,0,2068,1372,0,2650,880,0,3304,3538,0,1300,4288,0,4828,5110,0,
%U A370672 2464,6004,0,6640,2380,0,7654,3640,0
%N A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.
%C A370672 All solutions of this type can be found using a knight moving with some displacements dx and dy starting from some cell with coordinates (x,y): (x,y) -> (x+dx,y+dy) -> (x+2*dx,y+2*dy) -> ... -> (x,y) (all operations modulo n). For n <= 11 all solutions of n nonattacking queens on n X n a toroidal board problem are solutions of this type, for n >= 13 some solutions are not of this type (see A051906 for examples).
%H A370672 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2538">Arranging of N queens on toroidal board and generating pandiagonal Latin squares using them</a> (in Russian).
%H A370672 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2691">Numerical formula between number of cyclic diagonal Latin squares and number of toroidal n-queens problem solutions getting by knight movement</a> (in Russian).
%H A370672 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F A370672 a(n) = A123565(2*n+1) * (2*n+1).
%F A370672 a(n) = A338562(n) / (2n)!. - _Eduard I. Vatutin_, Mar 13 2024
%e A370672 For n=2*2+1=5 there are 10 solutions:
%e A370672 .
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%e A370672 | . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
%e A370672 | . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
%e A370672 | . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
%e A370672 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A370672 .
%e A370672 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A370672 | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
%e A370672 | . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
%e A370672 | . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
%e A370672 | . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
%e A370672 | Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
%e A370672 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A370672 .
%e A370672 so a(2)=10.
%Y A370672 Cf. A007705, A051906, A123565, A338562.
%K A370672 nonn
%O A370672 0,3
%A A370672 _Eduard I. Vatutin_, Feb 25 2024