This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A287695 #153 Dec 08 2023 20:40:14 %S A287695 1,0,0,1,1,0,3,824,614 %N A287695 Maximum number of diagonal Latin squares with the first row in ascending order that can be orthogonal to a given diagonal Latin square of order n. %C A287695 A Latin square is normalized if in the first row elements come in increasing order. Any diagonal Latin square orthogonal to a given one can be normalized by renaming its elements (which does not break diagonality and orthogonality). - _Max Alekseyev_, Dec 07 2019 %C A287695 For all orders n>3 there are diagonal Latin squares without orthogonal mates (also known as bachelor squares), so the minimum number of diagonal Latin squares that can be orthogonal to the same diagonal Latin square is zero. For order n=1 the single square is orthogonal to itself. For n=2 and n=3 diagonal Latin squares do not exist (see A274171). For n=6 orthogonal diagonal Latin squares do not exist (see A305571), so a(6)=0. - _Eduard I. Vatutin_, May 03 2021 %C A287695 a(n) >= A328873(n) - 1. - _Eduard I. Vatutin_, Mar 29 2021 %C A287695 a(10) >= 10 (Updated). - _Eduard I. Vatutin_, Apr 27 2018 %C A287695 a(11) >= 32462. - _Eduard I. Vatutin_ from T. Brada, Mar 11 2021 %C A287695 a(12) >= 3855983322. The result belongs to DLS, which has 30192 diagonal transversals. Calculations performed by a volunteer. - _Natalia Makarova_, _Tomáš Brada_, Nov 11 2021 %C A287695 a(13) >= 248703. - _Natalia Makarova_, _Tomáš Brada_, Apr 29 2021 %C A287695 a(14) >= 307662. - _Natalia Makarova_, Alex Chernov, Harry White, May 21 2021 %C A287695 a(16) >= 1658880, a(17) >= 2453352, a(18) >= 96, a(19) >= 1383, a(20) >= 995328, a(21) >= 995328, a(22) >= 432000, a(23) >= 525, a(24) >= 345600, a(25) >= 345600, a(26) >= 48, a(27) >= 345600, a(28) >= 663552, a(29) >= 663552, a(30) >= 40320. For values up to a(100), see the specified link "New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square". - _Natalia Makarova_, Alex Chernov, Harry White, Dec 06 2021 %H A287695 Natalia Makarova, <a href="https://boinc.progger.info/odlk/forum_thread.php?id=1&postid=1742#1742">Diagonal Latin square with 10 orthogonal squares</a> %H A287695 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=100&postid=938">DB CF ODLS of order 9</a> %H A287695 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133">Maximum number of normalized ODLS from one DLS</a> %H A287695 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=3218">Comments for result a(12) >= 3855983322</a> %H A287695 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=133&postid=2080">New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square</a> %H A287695 Eduard I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=87882#post87882">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian). %H A287695 Eduard I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=89479#post89479">Discussion about properties of diagonal Latin squares at forum.boinc.ru, square of order 9 with 516 orthogonal squares</a> (in Russian). %H A287695 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1576">About the A328873(N)-1 <= A287695(N) inequality between the maximum cardinality of clique and the maximum number of orthogonal normalized mates for one diagonal Latin square</a> (in Russian). %H A287695 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1734">About the diagonal Latin square of order 12 with 1764493860 orthogonal diagonal mates</a> (in Russian). %H A287695 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1664">Duplicate solutions removing using parallel and distributed DLX</a> (in Russian). %H A287695 Eduard I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_ls_cyclic_main_classes.pdf">Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares</a>, Recognition — 2021, pp. 77-79. (in Russian) %H A287695 Eduard I. Vatutin, <a href="/A287695/a287695_5.txt">Proving list (best known examples)</a>. %H A287695 Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina and Vitaly S. Titov, <a href="https://doi.org/10.25045/jpit.v10.i2.01">Central symmetry properties for diagonal Latin squares</a>, Problems of Information Technology (2019) No. 2, 3-8. %H A287695 Eduard I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk and V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_ls_odls_cnt_1_7.pdf">Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares</a>, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian). %H A287695 Eduard I. Vatutin, Natalia N. Nikitina and Maxim O. Manzuk, <a href="https://vk.com/wall162891802_1485">First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch</a> (in Russian). %H A287695 E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, <a href="http://evatutin.narod.ru/evatutin_spectra_t_dt_i_o_small_orders_thesis.pdf">On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order</a>, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian) %H A287695 E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, <a href="http://evatutin.narod.ru/evatutin_spectra_t_dt_i_o_high_orders_1.pdf">Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9</a> (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315. %H A287695 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>. %e A287695 From _Eduard I. Vatutin_, Mar 29 2021: (Start) %e A287695 One of the best existing diagonal Latin squares of order 7 %e A287695 0 1 2 3 4 5 6 %e A287695 2 3 1 5 6 4 0 %e A287695 5 6 4 0 1 2 3 %e A287695 4 0 6 2 3 1 5 %e A287695 6 2 0 1 5 3 4 %e A287695 1 5 3 4 0 6 2 %e A287695 3 4 5 6 2 0 1 %e A287695 has 3 orthogonal mates %e A287695 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 %e A287695 5 6 4 0 1 2 3 3 4 5 6 2 0 1 6 2 0 1 5 3 4 %e A287695 1 5 3 4 0 6 2 4 0 6 2 3 1 5 3 4 5 6 2 0 1 %e A287695 6 2 0 1 5 3 4 2 3 1 5 6 4 0 1 5 3 4 0 6 2 %e A287695 3 4 5 6 2 0 1 5 6 4 0 1 2 3 2 3 1 5 6 4 0 %e A287695 2 3 1 5 6 4 0 6 2 0 1 5 3 4 4 0 6 2 3 1 5 %e A287695 4 0 6 2 3 1 5 1 5 3 4 0 6 2 5 6 4 0 1 2 3 %e A287695 so a(7)=3. (End) %Y A287695 Cf. A001438, A274171, A305571, A328873. %K A287695 nonn,more,hard %O A287695 1,7 %A A287695 _Eduard I. Vatutin_, May 30 2017 %E A287695 Definition corrected by _Max Alekseyev_, Dec 07 2019 %E A287695 a(9) added by _Eduard I. Vatutin_, Dec 12 2020 %E A287695 Edited by _Max Alekseyev_, Apr 01 2022