# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a292517 Showing 1-1 of 1 %I A292517 #79 Aug 08 2023 22:22:44 %S A292517 48,495452160,38903149816763645952000, %T A292517 127654439655255918929515331054014121902080000 %N A292517 Number of doubly symmetric diagonal Latin squares of order 4n. %C A292517 A doubly symmetric square has symmetries in both horizontal and vertical planes. %C A292517 The plane symmetry requires one-to-one correspondence between the values of elements a[i,j] and a[N+1-i,j] in a vertical plane, and between the values of elements a[i,j] and a[i,N+1-j] in a horizontal plane for 1 <= i,j <= N. - _Eduard I. Vatutin_, Alexey D. Belyshev, Oct 09 2017 %C A292517 Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4). %C A292517 Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A293778(4n). - _Eduard I. Vatutin_, May 03 2021 %H A292517 A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian) %H A292517 Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian). %H A292517 Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian). %H A292517 E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79. %H A292517 Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8. %H A292517 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. 17-19. (in Russian) %H A292517 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. 30-36. (in Russian) %H A292517 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) %H A292517 Index entries for sequences related to Latin squares and rectangles %F A292517 a(n) = A287650(n) * (4n)!. %e A292517 Doubly symmetric diagonal Latin square example: %e A292517 0 1 2 3 4 5 6 7 %e A292517 3 2 7 6 1 0 5 4 %e A292517 2 3 1 0 7 6 4 5 %e A292517 6 7 5 4 3 2 0 1 %e A292517 7 6 3 2 5 4 1 0 %e A292517 4 5 0 1 6 7 2 3 %e A292517 5 4 6 7 0 1 3 2 %e A292517 1 0 4 5 2 3 7 6 %e A292517 In the horizontal direction there is a one-to-one correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4. %e A292517 In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5. %Y A292517 Cf. A003191, A287649, A287650, A293778, A340550. %K A292517 nonn,more,hard %O A292517 1,1 %A A292517 _Eduard I. Vatutin_, Sep 18 2017 %E A292517 a(2) corrected by _Eduard I. Vatutin_, Alexey D. Belyshev, Oct 09 2017 %E A292517 Edited and a(3) from A287650 added by _Max Alekseyev_, Aug 23 2018, Sep 07 2018 %E A292517 a(4) from _Andrew Howroyd_, May 31 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE