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a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.
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%I #45 Jan 23 2023 09:14:54

%S 1,0,0,1,2,1,3,31,99

%N a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

%C a(n) <= A287695(n) + 1.

%C a(n) <= A287764(n).

%C a(10) >= 10. It seems that a(10) = 10 due to long computational experiments within the Gerasim@Home volunteer distributed computing project did not reveal the existence of diagonal Latin squares of order 10 with the number of orthogonal diagonal Latin squares different from {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.

%C a(11) >= 112, a(12) >= 5079. - _Eduard I. Vatutin_, Nov 02 2021, updated Jan 23 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_1699">About the spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 9</a> (in Russian).

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1700">About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 10</a> (in Russian).

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1701">About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 11</a> (in Russian).

%H Eduard I. Vatutin, <a href="http://evatutin.narod.ru/spectra/spectra_dls_odls_all.png">Graphical representation of the spectra</a>.

%H 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 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 Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n1_1_item.txt">1</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n4_1_item.txt">4</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n5_2_items.txt">5</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n6_1_item.txt">6</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n7_3_items.txt">7</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n8_31_items.txt">8</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n9_99_items.txt">9</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n10_10_known_items.txt">10</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n11_112_known_items.txt">11</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_dls_odls_n12_xxxx_known_items.txt">12</a>).

%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 orthogonal diagonal Latin squares that a diagonal Latin square of order 7 may have is 0, 1, or 3. Since there are 3 distinct values, a(7)=3.

%Y Cf. A287695, A287764, A309344, A344105, A345370, A345760.

%K nonn,more,hard

%O 1,5

%A _Eduard I. Vatutin_, Jun 26 2021