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%I #55 Aug 16 2024 11:49:05
%S 0,0,0,1,2,1,21,61,64
%N a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.
%C a(n) <= A307164(n) - A307163(n) + 1.
%C a(n) <= A287764(n).
%C a(10) >= 98, a(11) >= 145, a(12) >= 259, a(13) >= 197, a(14) >= 362, a(15) >= 536, a(16) >= 789, a(17) >= 685, a(18) >= 419, a(19) >= 447, a(20) >= 1009, a(21) >= 740, a(22) >= 737, a(23) >= 885, a(24) >= 1610. - _Eduard I. Vatutin_, Oct 02 2021, updated May 21 2023
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1678">About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7</a> (in Russian).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1698">About the spectra of numerical characteristics of diagonal Latin squares of order 8</a> (in Russian).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1708">About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9</a> (in Russian).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2100">About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch</a> (in Russian).
%H Eduard I. Vatutin, <a href="http://evatutin.narod.ru/spectra/spectra_dls_intercalates_all.png">Graphical representation of the spectra</a>.
%H Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n1_0_items.txt">1</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n4_1_item.txt">4</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n5_2_items.txt">5</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n6_1_item.txt">6</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n7_21_items.txt">7</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n8_61_items.txt">8</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n9_64_items.txt">9</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n10_98_known_items.txt">10</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n11_145_known_items.txt">11</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n12_259_known_items.txt">12</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n13_197_known_items.txt">13</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n14_362_known_items.txt">14</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n15_536_known_items.txt">15</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n16_789_known_items.txt">16</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n17_685_known_items.txt">17</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n18_419_known_items.txt">18</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n19_447_known_items.txt">19</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n20_1009_known_items.txt">20</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n21_740_known_items.txt">21</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n22_737_known_items.txt">22</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n23_883_known_items.txt">23</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_intercalates_n24_1610_known_items.txt">24</a>).
%H E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, <a href="http://evatutin.narod.ru/evatutin_dls_heur_spectra_method_2.pdf">Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares</a>, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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 E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, <a href="http://evatutin.narod.ru/evatutin_dls_heur_spectra_method.pdf">Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares</a>, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
%H <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e For n=7 the number of intercalates that a diagonal Latin square of order 7 may have is 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26, or 30. Since there are 21 distinct values, a(7)=21.
%Y Cf. A287764, A307163, A307164, A309344, A344105, A345370, A345761.
%K nonn,more,hard
%O 1,5
%A _Eduard I. Vatutin_, Jun 26 2021
%E a(9) added by _Eduard I. Vatutin_, Oct 22 2022