login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A343867 Number of semicyclic pandiagonal Latin squares of order 2*n+1 with the first row in ascending order. 5
0, 0, 0, 0, 0, 0, 1560, 0, 34000, 175104, 0, 22417824, 313235960, 0, 83574857328, 1729671003296 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
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.
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.
FORMULA
a(n) >= 4*(A071607(n) - A123565(2*n+1)).
EXAMPLE
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
PROG
(PARI) \\ See Links
CROSSREFS
Cf. A071607, A123565 (cyclic), A338620, A343868.
Sequence in context: A362203 A368785 A368786 * A302057 A092001 A069475
KEYWORD
nonn,more
AUTHOR
Andrew Howroyd, May 08 2021
EXTENSIONS
a(12)-a(15) from Jim White, Aug 03 2021
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)