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A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

%I A345370 #73 Mar 02 2025 07:59:34
%S A345370 1,0,0,1,2,2,14,47,182
%N A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.
%C A345370 a(n) <= A287648(n) - A287647(n) + 1.
%C A345370 a(n) <= A287764(n).
%C A345370 Conjecture: a(12) = A287648(12) - A287647(12) + 1. - _Natalia Makarova_, Oct 26 2021
%C A345370 a(10) >= 736, a(11) >= 1344, a(12) >= 17693, a(13) >= 18241, a(14) >= 294053, a(15) >= 1958394, a(16) >= 13715. - _Eduard I. Vatutin_, Oct 29 2021, updated Mar 02 2025
%H A345370 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 A345370 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 A345370 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1790">On the falsity of conjecture that spectra of diagonal transversals for diagonal Latin squares of order 12 is solid</a> (in Russian).
%H A345370 Eduard I. Vatutin, <a href="http://evatutin.narod.ru/spectra/spectra_dls_diagonal_transversals_all.png">Graphical representation of the spectra</a>.
%H A345370 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 A345370 Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n1_1_item.txt">1</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n4_1_item.txt">4</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n5_2_items.txt">5</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n6_2_items.txt">6</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n7_14_items.txt">7</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n8_47_items.txt">8</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n9_182_items.txt">9</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n10_736_known_items.txt">10</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n11_1344_known_items.txt">11</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n12_17693_known_items.txt">12</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_diagonal_transversals_n13_18241_known_items.txt">13</a>).
%H A345370 E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_ls_distr_diag.pdf">Distributed diagonalization strategy for Latin squares</a>, Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2023. pp. 309-311. (in Russian)
%H A345370 E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, 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 A345370 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 A345370 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 A345370 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 A345370 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A345370 For n=7 the number of diagonal transversals that a diagonal Latin square of order 7 may have is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, or 27. Since there are 14 distinct values, a(7)=14.
%Y A345370 Cf. A287647, A287648, A309344, A344105, A345760, A345761, A349199.
%K A345370 nonn,more,hard
%O A345370 1,5
%A A345370 _Eduard I. Vatutin_, Jun 16 2021
%E A345370 a(8) added by _Eduard I. Vatutin_, Jul 15 2021
%E A345370 a(9) added by _Eduard I. Vatutin_, Oct 20 2022