%I #62 Oct 06 2023 17:59:55

%S 0,0,0,12,4,9,30,112,72

%N Maximum number of intercalates in a diagonal Latin square of order n.

%C An intercalate is a 2 X 2 subsquare of a Latin square.

%C 0 <= A307163(n) <= A307164(n) <= A092237(n). - _Eduard I. Vatutin_, Sep 21 2020

%C a(10) >= 101, a(11) >= 94, a(12) >= 252, a(13) >= 156, a(14) >= 353. - _Eduard I. Vatutin_, May 31 2021, updated Sep 10 2023

%H Eduard I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&amp;m=92687#post92687">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian).

%H 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 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 Eduard I. Vatutin, <a href="/A307164/a307164_3.txt">Proving list (best known examples)</a>.

%H 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 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 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 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 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.

%e From _Eduard I. Vatutin_, May 31 2021: (Start)

%e One of the best known diagonal Latin squares of order n=5

%e 0 1 2 3 4

%e 4 2 0 1 3

%e 1 4 3 2 0

%e 3 0 1 4 2

%e 2 3 4 0 1

%e has 4 intercalates:

%e . . 2 3 . . . . . . . . . . . . . . . .

%e . . . . . . . 0 . 3 . . . . . . . . . .

%e . . 3 2 . . . 3 . 0 1 . 3 . . . 4 3 . .

%e . . . . . . . . . . 3 . 1 . . . . . . .

%e . . . . . . . . . . . . . . . . 3 4 . .

%e so a(5)=4. (End)

%Y Cf. A092237, A307163, A345760.

%K nonn,more,hard

%O 1,4

%A _Eduard I. Vatutin_, Mar 27 2019

%E a(9) added by _Eduard I. Vatutin_, Sep 21 2020