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Polya proved (see Ahrens) that the number of solutions to this problem for an m X m board is > 0 iff m is coprime to 6. - Jonathan Vos Post, Feb 20 2005

The sequence has been computed up to n = 23 by Rivin, Vardi & Zimmermann, see p. 637 of their paper from 1994. Further terms were calculated by the submitter, Dec 17 1999 and Jan 11 2001.

a(n) is divisible by n.

Only terms indexed by odd numbers coprime to 3 are nonzero, therefore A007705(n) = a(2n+1) is the main entry. - M. F. Hasler, Jul 01 2019

From Eduard I. Vatutin, Nov 27 2023: (Start)

For n <= 11 all solutions 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 >= 13 some solutions are same, but not all (see examples).

All solutions of n-queens problem on toroidal chessboard are also solutions of n-queens problem on classical chessboard; the converse is not true, so a(n) <= A000170(n).

(End)

A complete mapping of a cyclic group (Z_n,+) is a permutation f(x) of Z_n such that f(0)=0 and such that f(x)-x is also a permutation.

a(n)=TSQ(n)/n where TSQ(n) is the number of solutions of the toroidal semi-n-queen problem (A006717 is the sequence TSQ(2k-1)).

Stated another way, this is the number of "good" permutations on 2n+1 elements (see A006717) that start with 0. [Novakovich]. - N. J. A. Sloane, Feb 22 2011

Horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). Vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i). Cyclic diagonal Latin squares (see A338562) fall under the definition of vertically and horizontally semicyclic diagonal Latin squares simultaneously, in this type of squares each row r(i) is obtained from the previous one r(i-1) using cyclic shift by some value d. Definition from A343867 includes this type of squares but not only it.

Pandiagonal Latin squares exist only for odd orders not divisible by 3. All pandiagonal Latin squares for orders less than 13 are cyclic which are not counted by this sequence.

Semicyclic Latin squares are defined in the Atkin reference where the first nonzero term of this sequence is given. They are cyclic in a single direction. The direction can be horizontal or vertical or any other step such as a knights move.

Each symbol in a semicyclic Latin square occupies the same pattern of squares up to translation on the torus which in the case of a pandiagonal square is a solution to the toroidal n-queens problem.

For prime 2n+1, a(n) is a multiple of 2n+1.

A pandiagonal Latin square is a Latin square in which the diagonal, antidiagonal and all broken diagonals and antidiagonals are transversals.

For orders n = 5, 7 and 11 all pandiagonal Latin squares are cyclic, so a(n) = A123565(2n+1) for n < 6. For n=6 (order 13), this is not true and there are 12386 inequivalent squares; of these 10 are cyclic (in all directions) and 1560 are semi-cyclic (A343867).

Pandiagonal Latin squares exist only for odd orders not divisible by 3. This is because the positions of each symbol are a solution to the toroidal n-queens problem which only has solutions for these sizes. - Andrew Howroyd, May 26 2021

See Comments in A372922.

A horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example).

A horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). Similarly, a vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i).

A complete mapping of a cyclic group (Z_m,+) is a permutation f(x) of Z_m with f(0)=0 such that f(x)-x is also a permutation. a(n) is the number of complete mappings f(x) of the cyclic group Z_{2n+1} such that f^(-1)=f.

In other words, a(n) is the number of complete mappings fixed under the reflection operator R, where R(f)=f^(-1). Reflection R is not only a symmetry operator of complete mappings, but also one of the (Toroidal)-(semi) N-Queen problems and of the strong complete mappings problem.