

A292517


Number of doubly symmetric diagonal Latin squares of order 4n.


4




OFFSET

1,1


COMMENTS

Doubly symmetric square has symmetries both in horizontal and vertical planes.
The plane symmetry requires onetoone correspondence between the values of elements a[i][j] and a[Ni][j] in a vertical plane, and between the values of elements a[i][j] and a[i][Nj] in a horizontal plane for 1 <= i,j <= N.  Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4).


LINKS

Table of n, a(n) for n=1..3.
A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 7479.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 1719 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 3036 (in Russian)
Index entries for sequences related to Latin squares and rectangles


FORMULA

a(n) = A287650(n) * (4n)!.


EXAMPLE

Doubly symmetric diagonal Latin square example:
0 1 2 3 4 5 6 7
3 2 7 6 1 0 5 4
2 3 1 0 7 6 4 5
6 7 5 4 3 2 0 1
7 6 3 2 5 4 1 0
4 5 0 1 6 7 2 3
5 4 6 7 0 1 3 2
1 0 4 5 2 3 7 6
In the horizontal direction there is a onetoone correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4.
In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5.


CROSSREFS

Cf. A003191, A287649, A287650.
Sequence in context: A165643 A165047 A291865 * A272096 A115480 A214953
Adjacent sequences: A292514 A292515 A292516 * A292518 A292519 A292520


KEYWORD

bref,nonn,more


AUTHOR

Eduard I. Vatutin, Sep 18 2017


EXTENSIONS

a(2) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Edited and a(3) from A287650 added by Max Alekseyev, Aug 23 2018, Sep 07 2018


STATUS

approved



