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 A307164 #70 Mar 12 2025 16:15:02 %S A307164 0,0,0,12,4,9,30,112,72 %N A307164 Maximum number of intercalates in a diagonal Latin square of order n. %C A307164 An intercalate is a 2 X 2 subsquare of a Latin square. %C A307164 0 <= A307163(n) <= A307164(n) <= A092237(n). - _Eduard I. Vatutin_, Sep 21 2020 %C A307164 a(10) >= 109, a(11) >= 172, a(12) >= 324, a(13) >= 180, a(14) >= 391, a(15) >= 630, a(16) >= 960, a(17) >= 736, a(18) >= 547, a(19) >= 457, a(20) >= 1100, a(21) >= 785, a(22) >= 887, a(23) >= 899, a(24) >= 1680, a(25) >= 1700, a(26) >= 1299, a(27) >= 1372, a(28) >= 2892. - _Eduard I. Vatutin_, May 31 2021, updated Mar 02 2025 %C A307164 If, in theory, all unordered pairs of rows and columns form intercalate in their intersection, total number of intercalates will be (n*(n-1))^2, so a(n) <= (n*(n-1))^2, a(n) is asymptotically less than O(n^4). In practice a(n) << (n*(n-1))^2. - _Eduard I. Vatutin_, Mar 05 2025 %H A307164 Eduard I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=92687#post92687">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian). %H A307164 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1346">About the maximum number of intercalates in a diagonal Latin squares of order 9</a> (in Russian). %H A307164 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2475">About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14</a> (in Russian). %H A307164 Eduard I. Vatutin, <a href="/A307164/a307164_4.txt">Proving list (best known examples)</a>. %H A307164 Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, <a href="https://doi.org/10.1007/978-3-030-66895-2_9">Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10</a>, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer, Cham (2020), 127-146. %H A307164 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 A307164 Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Alexandr M. Albertyan, and Ilya 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 A307164 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 A307164 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>. %e A307164 From _Eduard I. Vatutin_, May 31 2021: (Start) %e A307164 One of the best known diagonal Latin squares of order n=5 %e A307164 0 1 2 3 4 %e A307164 4 2 0 1 3 %e A307164 1 4 3 2 0 %e A307164 3 0 1 4 2 %e A307164 2 3 4 0 1 %e A307164 has 4 intercalates: %e A307164 . . 2 3 . . . . . . . . . . . . . . . . %e A307164 . . . . . . . 0 . 3 . . . . . . . . . . %e A307164 . . 3 2 . . . 3 . 0 1 . 3 . . . 4 3 . . %e A307164 . . . . . . . . . . 3 . 1 . . . . . . . %e A307164 . . . . . . . . . . . . . . . . 3 4 . . %e A307164 so a(5)=4. (End) %Y A307164 Cf. A092237, A307163, A345760. %K A307164 nonn,more,hard %O A307164 1,4 %A A307164 _Eduard I. Vatutin_, Mar 27 2019 %E A307164 a(9) added by _Eduard I. Vatutin_, Sep 21 2020