A338562 Number of cyclic diagonal Latin squares of order 2n+1.

1, 0, 240, 20160, 0, 319334400, 62270208000, 0, 4979623993344000, 1946321606541312000, 0, 517040334777699532800000, 155112100433309859840000000, 0, 229885811837232250818134016000000, 230239482316981838896315760640000000, 0, 82665183731089159437333210700185600000000

Offset: 0

Author

Eduard I. Vatutin, Nov 02 2020

Keywords

• nonn
• easy

Comments

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places.

Equivalently, a Latin square is cyclic if and only if each row is a cyclic permutation of the first row and each column is a cyclic permutation of the first column.

Every cyclic diagonal Latin square is a cyclic Latin square, so a(n) <= A338522(2*n+1).

Cyclic diagonal Latin squares exist only for odd orders not divisible by 3. - Andrew Howroyd, May 26 2021

Examples

			For n=3 there are 6 cyclic Latin squares of order 7 with the first row in ascending order, only 4 of them are diagonal:
  0 1 2 3 4 5 6   0 1 2 3 4 5 6   0 1 2 3 4 5 6   0 1 2 3 4 5 6
  2 3 4 5 6 0 1   3 4 5 6 0 1 2   4 5 6 0 1 2 3   5 6 0 1 2 3 4
  4 5 6 0 1 2 3   6 0 1 2 3 4 5   1 2 3 4 5 6 0   3 4 5 6 0 1 2
  6 0 1 2 3 4 5   2 3 4 5 6 0 1   5 6 0 1 2 3 4   1 2 3 4 5 6 0
  1 2 3 4 5 6 0   5 6 0 1 2 3 4   2 3 4 5 6 0 1   6 0 1 2 3 4 5
  3 4 5 6 0 1 2   1 2 3 4 5 6 0   6 0 1 2 3 4 5   4 5 6 0 1 2 3
  5 6 0 1 2 3 4   4 5 6 0 1 2 3   3 4 5 6 0 1 2   2 3 4 5 6 0 1
and 4*7! = 20160 cyclic diagonal Latin squares.
		

Links

• Eduard I. Vatutin, Enumerating cyclic Latin squares and Euler totient function calculating using them, High-performance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 40-48. (in Russian)
• Eduard I. Vatutin, Numerical formula between number of cyclic diagonal Latin squares and number of toroidal n-queens problem solutions getting by knight movement (in Russian).
• E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A123565 (ordered first row), A338522, A341585 (main classes), A342306, A370672.

Programs

PARI

a(n)={my(m=2*n+1); m!*if(gcd(m, 6)==1, sum(k=1, m, gcd(k^3-k, m)==1))} \\ Andrew Howroyd, Apr 30 2021

Formula

a(n) = A123565(2*n+1) * (2*n+1)!.

a(n) = A370672(n) * (2n)!. - Eduard I. Vatutin, Mar 13 2024

Extensions

More terms from Andrew Howroyd, Apr 30 2021

Zero terms for even orders removed by Andrew Howroyd, May 26 2021