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A349199 a(n) is the number of distinct numbers of diagonal transversals that an orthogonal diagonal Latin square of order n can have.

%I A349199 #22 Jan 29 2023 09:43:27
%S A349199 1,0,0,1,1,0,3,31,165
%N A349199 a(n) is the number of distinct numbers of diagonal transversals that an orthogonal diagonal Latin square of order n can have.
%C A349199 An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A345370(n).
%C A349199 a(10) >= 390, a(11) >= 560, a(12) >= 13429. - _Eduard I. Vatutin_, Nov 10 2021, updated Jan 29 2023
%H A349199 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1709">About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11</a> (in Russian).
%H A349199 Eduard I. Vatutin, <a href="http://evatutin.narod.ru/spectra/spectra_odls_diagonal_transversals_all.png">Graphical representation of the spectra</a>.
%H A349199 Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n1_1_item.txt">1</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n4_1_item.txt">4</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n5_1_item.txt">5</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n7_3_items.txt">7</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n8_31_items.txt">8</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n9_165_items.txt">9</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n10_390_known_items.txt">10</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n11_560_known_items.txt">11</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_odls_diagonal_transversals_n12_13429_known_items.txt">12</a>).
%H A349199 E. I. Vatutin, 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_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 A349199 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A349199 For n=8 the number of diagonal transversals that an orthogonal diagonal Latin square of order 8 may have is 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 32, 36, 38, 40, 42, 44, 48, 52, 56, 64, 72, 88, 96, or 120. Since there are 31 distinct values, a(8)=31.
%Y A349199 Cf. A305570, A345370.
%K A349199 nonn,more,hard
%O A349199 1,7
%A A349199 _Eduard I. Vatutin_, Nov 10 2021