Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n, https://oeis.org/A360220 n=1, a(1)=1 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: trivial 0 n=2, a(2)=0 - n=3, a(3)=0 - n=4, a(4)=4 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 3 2 1 0 1 0 3 2 2 3 0 1 n=5, a(5)=5 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 4 2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2 n=6, a(6)=0 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX (for n=6 ODLS does not exist) - n=7, a(7)=27 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 4 5 6 2 3 1 5 6 4 0 5 6 4 0 1 2 3 4 0 6 2 3 1 5 6 2 0 1 5 3 4 1 5 3 4 0 6 2 3 4 5 6 2 0 1 n=8, a(8)=120 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 2 3 0 1 6 7 4 5 1 5 4 0 7 3 2 6 5 4 7 6 1 0 3 2 3 7 6 2 5 1 0 4 7 6 5 4 3 2 1 0 4 0 1 5 2 6 7 3 6 2 3 7 0 4 5 1 n=9, a(9)=333 Article: E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. Announcement 1: https://vk.com/wall162891802_1368, Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Sep 26 2020 (a(9)>=333) Announcement 2: https://vk.com/wall162891802_1485, Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Dec 07 2020 (a(9)=333) Way of finding: brute force using X-based fillings, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 2 4 3 0 7 6 8 1 5 5 0 6 8 2 1 4 3 7 1 5 4 7 6 0 2 8 3 6 2 8 5 3 4 7 0 1 3 7 0 2 1 8 5 4 6 8 3 5 6 0 7 1 2 4 4 6 7 1 8 2 3 5 0 7 8 1 4 5 3 0 6 2 n=10, a(10)>=866 Announcement: J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, and W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics. 1992. Vol. 139. pp. 43-49. Way of finding: horizontally symmetric string inverse square, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 1 2 3 4 0 9 5 6 7 8 3 5 9 1 2 7 8 0 4 6 5 9 8 7 6 3 2 1 0 4 7 6 4 9 8 1 0 5 3 2 4 0 1 2 3 6 7 8 9 5 9 8 7 6 5 4 3 2 1 0 2 3 5 0 1 8 9 4 6 7 6 4 0 8 7 2 1 9 5 3 8 7 6 5 9 0 4 3 2 1 n=11, a(11)>=4828 Announcement: https://boinc.tbrada.eu/forum_thread.php?id=3104&postid=4149, Tomas Brada, Jul 09 2020 Way of finding: combinatorial structure from cyclic squares, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 10 1 2 3 5 6 9 7 10 0 4 8 7 10 8 0 3 1 5 9 6 2 4 3 5 9 4 10 6 8 0 2 7 1 4 6 7 10 1 8 2 3 9 0 5 5 9 4 6 8 7 0 1 3 10 2 10 8 0 1 5 2 9 4 7 3 6 6 7 10 8 2 0 3 5 4 1 9 8 0 1 2 9 3 4 6 10 5 7 2 3 5 9 7 4 10 8 1 6 0 9 4 6 7 0 10 1 2 5 8 3 n=12, a(12)>=30192 Announcement: https://vk.com/wall162891802_1732, Eduard I. Vatutin, Aug 18 2021 Way of finding: diagonalizing of known double Brown square (obtained using the composite squares method + diagonalizing), Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 9 8 11 5 10 0 6 7 5 8 10 6 11 4 1 3 9 7 0 2 11 7 5 8 10 2 9 1 3 6 4 0 7 5 8 10 6 3 0 2 4 11 9 1 9 0 1 2 3 7 10 11 5 4 8 6 6 11 7 5 8 1 4 0 2 10 3 9 10 6 11 7 5 0 3 9 1 8 2 4 3 4 9 0 1 6 5 10 11 2 7 8 2 3 4 9 0 10 7 8 6 1 11 5 4 9 0 1 2 11 8 6 7 3 5 10 8 10 6 11 7 9 2 4 0 5 1 3 n=13, a(13)>=131106 Announcement: https://boinc.tbrada.eu/forum_thread.php?id=3104&postid=4133, Tomas Brada, Jul 03 2020 Way of finding: cyclic squares 0 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 0 1 4 5 6 7 8 9 10 11 12 0 1 2 3 6 7 8 9 10 11 12 0 1 2 3 4 5 8 9 10 11 12 0 1 2 3 4 5 6 7 10 11 12 0 1 2 3 4 5 6 7 8 9 12 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 0 3 4 5 6 7 8 9 10 11 12 0 1 2 5 6 7 8 9 10 11 12 0 1 2 3 4 7 8 9 10 11 12 0 1 2 3 4 5 6 9 10 11 12 0 1 2 3 4 5 6 7 8 11 12 0 1 2 3 4 5 6 7 8 9 10 n=17, a(17)>=204995269 Announcement: https://vk.com/wall162891802_1412, Eduard I. Vatutin, Oct 28 2020 Way of finding: cyclic square 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n=19, a(19)>=11254190082 Announcement: https://vk.com/wall162891802_1412, Eduard I. Vatutin, Oct 28 2020 Way of finding: cyclic square 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 9 10 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 11 12 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 13 14 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 15 16 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Jul 19 2022