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 A292517 Number of doubly symmetric diagonal Latin squares of order 4n. 4
 48, 495452160, 38903149816763645952000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Doubly symmetric square has symmetries both in horizontal and vertical planes. The plane symmetry requires one-to-one correspondence between the values of elements a[i][j] and a[N-i][j] in a vertical plane, and between the values of elements a[i][j] and a[i][N-j] 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 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. 74-79. 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) 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) 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 one-to-one 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

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Last modified September 25 20:53 EDT 2018. Contains 315425 sequences. (Running on oeis4.)