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 A328873 #127 Jun 25 2025 11:38:36 %S A328873 1,0,0,2,2,1,4,6,6 %N A328873 Maximal size of a set of pairwise mutually orthogonal diagonal Latin squares of order n. %C A328873 From _Andrew Howroyd_, Nov 08 2019: (Start) %C A328873 A diagonal Latin square of order n is an n X n array with every integer from 0 to n-1 in every row, every column, and both main diagonals. %C A328873 Of course if even one example exists, then a(n) >= 1. %C A328873 A274806 gives the number of diagonal Latin squares and A274806(6) is nonzero. This suggests that although it is not possible to have a pair of orthogonal diagonal Latin squares, a(6) should be 1 here. (End) %C A328873 a(1) = 1 because there is only one (trivial) diagonal Latin square of order 1. It is orthogonal to itself, so if we allow the consideration of multiple copies of the same diagonal Latin square, we get a(1) = infinity instead. %C A328873 From _Eduard I. Vatutin_, Mar 27 2021: (Start) %C A328873 a(n) <= A287695(n) + 1. %C A328873 a(p) >= A123565(p) = p-3 for all odd prime p due to existence of clique from cyclic MODLS of order p with at least A123565(p) items. It seems that for some orders p clique from cyclic MODLS can be extended by adding none cyclic DLS that are orthogonal to all cyclic DLS. (End) %C A328873 a(n) <= A001438(n). - _Max Alekseyev_, Nov 08 2019 %C A328873 a(10) >= 2; a(11) >= 8; a(12) >= 4; a(13) >= 10; a(14) >= 2; a(15) >= 4. - _Natalia Makarova_, Sep 03 2020; updated May 30 2021 %C A328873 a(16) >= 14, a(17) >= 14, a(18) >= 2, a(19) >= 16, a(20) >= 2. - _Natalia Makarova_, Jan 08 2021 %H A328873 R. J. R. Abel, Charles J. Colbourn, and Jeffrey H. Dinitz, <a href="https://www.researchgate.net/publication/329786731_Mutually_orthogonal_latin_squares_MOLS">Mutually Orthogonal Latin Squares (MOLS)</a> [Note the first author, Julian Abel, has the initials R. J. R. A. - _N. J. A. Sloane_, Nov 05 2020] %H A328873 B. Du, <a href="https://ajc.maths.uq.edu.au/pdf/7/ocr-ajc-v7-p87.pdf">New Bounds For Pairwise Orthogonal Diagonal Latin Squares</a>, Australasian Journal of Combinatorics 7 (1993), pp.87-99. %H A328873 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=115">MODLS of order 15</a> %H A328873 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=117">Complete MOLS systems</a> %H A328873 Natalia Makarova, <a href="http://www.natalimak1.narod.ru/diagon.htm">Orthogonal Diagonal Latin squares</a> %H A328873 Natalia Makarova, <a href="/A328873/a328873.txt">Mutually Orthogonal Diagonal Latin squares (MODLS) for orders 9 - 20</a> %H A328873 Natalia Makarova, <a href="https://boinc.multi-pool.info/latinsquares/forum_thread.php?id=120">MOLS and MODLS of order 12</a> %H A328873 E. I. Vatutin, <a href="https://vk.com/wall162891802_980">Discussion about properties of diagonal Latin squares</a> (in Russian), Oct 29 2019. %H A328873 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1492">On the falsity of Makarova's proof that a(9) = 6</a> (in Russian). %H A328873 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1497">About the cliques from orthogonal diagonal Latin squares of order 9, brute force based proof that a(9) = 6</a> (in Russian). %H A328873 E. I. Vatutin, M. O. Manzuk, V. S. Titov, S. E. Kochemazov, A. D. Belyshev, N. N. Nikitina, <a href="http://evatutin.narod.ru/evatutin_ls_all_structs_n1to8_art.pdf">Orthogonality-based classification of diagonal latin squares of orders 1-8</a>, High-performance computing systems and technologies. Vol. 3. No. 1. 2019. pp. 94-100. (in Russian). %H A328873 E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, O. S. Zaikin, A. D. Belyshev, <a href="http://evatutin.narod.ru/evatutin_dls_cliques_properties.pdf">Cliques properties from diagonal Latin squares of small order</a>, Intellectual and Information Systems (Intellect - 2019). Tula, 2019. pp. 17-23. (in Russian). %H A328873 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 A328873 Eduard I. Vatutin, <a href="/A328873/a328873_2.txt">Proving list (best known examples)</a>. %H A328873 Wikipedia, <a href="https://en.wikipedia.org/wiki/Clique_problem">Clique problem</a>. %H A328873 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>. %e A328873 Orthogonal pair of Diagonal Latin squares of order 18: %e A328873 1 5 15 16 17 18 2 14 4 13 3 7 12 10 8 6 11 9 %e A328873 8 2 6 15 16 17 18 1 5 14 4 13 11 9 7 12 10 3 %e A328873 14 9 3 7 15 16 17 2 6 1 5 12 10 8 13 11 4 18 %e A328873 13 1 10 4 8 15 16 3 7 2 6 11 9 14 12 5 18 17 %e A328873 12 14 2 11 5 9 15 4 8 3 7 10 1 13 6 18 17 16 %e A328873 11 13 1 3 12 6 10 5 9 4 8 2 14 7 18 17 16 15 %e A328873 3 12 14 2 4 13 7 6 10 5 9 1 8 18 17 16 15 11 %e A328873 9 10 11 12 13 14 1 15 16 17 18 8 7 6 5 4 3 2 %e A328873 6 7 8 9 10 11 12 18 17 16 15 5 4 3 2 1 14 13 %e A328873 5 6 7 8 9 10 11 16 15 18 17 4 3 2 1 14 13 12 %e A328873 7 8 9 10 11 12 13 17 18 15 16 6 5 4 3 2 1 14 %e A328873 4 15 16 17 18 1 8 13 3 12 2 14 6 11 9 7 5 10 %e A328873 15 16 17 18 14 7 9 12 2 11 1 3 13 5 10 8 6 4 %e A328873 16 17 18 13 6 8 3 11 1 10 14 15 2 12 4 9 7 5 %e A328873 17 18 12 5 7 2 4 10 14 9 13 16 15 1 11 3 8 6 %e A328873 18 11 4 6 1 3 5 9 13 8 12 17 16 15 14 10 2 7 %e A328873 10 3 5 14 2 4 6 8 12 7 11 18 17 16 15 13 9 1 %e A328873 2 4 13 1 3 5 14 7 11 6 10 9 18 17 16 15 12 8 %e A328873 and %e A328873 1 8 14 13 12 11 3 9 6 5 7 4 15 16 17 18 10 2 %e A328873 5 2 9 1 14 13 12 10 7 6 8 15 16 17 18 11 3 4 %e A328873 15 6 3 10 2 1 14 11 8 7 9 16 17 18 12 4 5 13 %e A328873 16 15 7 4 11 3 2 12 9 8 10 17 18 13 5 6 14 1 %e A328873 17 16 15 8 5 12 4 13 10 9 11 18 14 6 7 1 2 3 %e A328873 18 17 16 15 9 6 13 14 11 10 12 1 7 8 2 3 4 5 %e A328873 2 18 17 16 15 10 7 1 12 11 13 8 9 3 4 5 6 14 %e A328873 14 1 2 3 4 5 6 15 16 17 18 13 12 11 10 9 8 7 %e A328873 4 5 6 7 8 9 10 17 18 15 16 3 2 1 14 13 12 11 %e A328873 13 14 1 2 3 4 5 18 17 16 15 12 11 10 9 8 7 6 %e A328873 3 4 5 6 7 8 9 16 15 18 17 2 1 14 13 12 11 10 %e A328873 7 13 12 11 10 2 1 8 5 4 6 14 3 15 16 17 18 9 %e A328873 12 11 10 9 1 14 8 7 4 3 5 6 13 2 15 16 17 18 %e A328873 10 9 8 14 13 7 18 6 3 2 4 11 5 12 1 15 16 17 %e A328873 8 7 13 12 6 18 17 5 2 1 3 9 10 4 11 14 15 16 %e A328873 6 12 11 5 18 17 16 4 1 14 2 7 8 9 3 10 13 15 %e A328873 11 10 4 18 17 16 15 3 14 13 1 5 6 7 8 2 9 12 %e A328873 9 3 18 17 16 15 11 2 13 12 14 10 4 5 6 7 1 8 %e A328873 so a(18) >= 2. %Y A328873 Cf. A001438, A274806, A287695. %K A328873 nonn,more,hard %O A328873 1,4 %A A328873 _Eduard I. Vatutin_, Oct 29 2019 %E A328873 a(6) corrected by _Max Alekseyev_ and _Andrew Howroyd_, Nov 08 2019 %E A328873 a(9) added by _Eduard I. Vatutin_, Feb 02 2021