A343867 Number of semicyclic pandiagonal Latin squares of order 2*n+1 with the first row in ascending order.

0, 0, 0, 0, 0, 0, 1560, 0, 34000, 175104, 0, 22417824, 313235960, 0, 83574857328, 1729671003296

Offset: 0

Author

Andrew Howroyd, May 08 2021

Keywords

• nonn
• more

Comments

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.

Examples

			The following is an example of an order 13 semicyclic square with a step of (1,4). This means moving down one row and across by 4 columns increases the cell value by 1 modulo 13. Symbols can be relabeled to give a square with the first row in ascending order.
   0 11  1  7  5  9  3 10  4  8  6 12  2
   9  7  0  3  1 12  2  8  6 10  4 11  5
  11  5 12  6 10  8  1  4  2  0  3  9  7
   1  4 10  8 12  6  0  7 11  9  2  5  3
  10  3  6  4  2  5 11  9  0  7  1  8 12
   8  2  9  0 11  4  7  5  3  6 12 10  1
   7  0 11  2  9  3 10  1 12  5  8  6  4
   6  9  7  5  8  1 12  3 10  4 11  2  0
   5 12  3  1  7 10  8  6  9  2  0  4 11
   3  1  5 12  6  0  4  2  8 11  9  7 10
  12 10  8 11  4  2  6  0  7  1  5  3  9
   2  6  4 10  0 11  9 12  5  3  7  1  8
   4  8  2  9  3  7  5 11  1 12 10  0  6
...
a(12) = 4*(A071607(12) - A123565(25)) + 11240. - _Jim White_, Jul 22 2021
a(14) = 4*(A071607(14) - A123565(29)) + 91176. - _Jim White_, Jul 24 2021
a(15) = 4*(A071607(15) - A123565(31)) + 334800. - _Jim White_, Aug 03 2021
		

Links

• A. O. L. Atkin, L. Hay, and R. G. Larson, Enumeration and construction of pandiagonal Latin squares of prime order, Computers & Mathematics with Applications, Volume. 9, Iss. 2, 1983, pp. 267-292.
• Andrew Howroyd, PARI Program for Initial Terms.
• Natalia Makarova from Harry White, 1560 semi-cyclic Latin squares of order 13.
• Natalia Makarova from Harry White, 34000 semi-cyclic Latin squares of order 17.
• Eduard I. Vatutin, 175104 semi-cyclic Latin squares of order 19.

Crossrefs

Cf. A071607, A123565 (cyclic), A338620, A343868.

Programs

PARI

\\ See Links

Formula

a(n) >= 4*(A071607(n) - A123565(2*n+1)).

Extensions

a(12)-a(15) from Jim White, Aug 03 2021