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A366333 a(n) is the number of distinct numbers of diagonal transversals that a semicyclic diagonal Latin square of order 2n+1 can have.

%I A366333 #11 Dec 15 2023 15:54:52
%S A366333 1,0,1,1,0,2,20,0,271,1208,0
%N A366333 a(n) is the number of distinct numbers of diagonal transversals that a semicyclic diagonal Latin square of order 2n+1 can have.
%C A366333 A horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). A vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i). Cyclic diagonal Latin squares (see A338562) fall under the definition of vertically and horizontally semicyclic diagonal Latin squares simultaneously, in this type of squares each row r(i) is obtained from the previous one r(i-1) using cyclic shift by some value d.
%C A366333 Semicyclic diagonal Latin squares do not exist for even orders n.
%H A366333 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2443">About the spectra of numerical characteristics of different types of cyclic diagonal Latin squares</a> (in Russian).
%H A366333 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2450">About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 17</a> (in Russian).
%H A366333 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2453">About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 19</a> (in Russian).
%H A366333 Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n1_1_item.txt">1</a>, <a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n5_1_item.txt">5</a>, <a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n7_1_item.txt">7</a>, <a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n11_2_items.txt">11</a>, <a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n13_20_items.txt">13</a>, <a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n17_271_items.txt">17</a>, <a href="http://evatutin.narod.ru/spectra/sp_horz_semicyclic_dls_diagonal_transversals_n19_1208_items.txt">19</a>).
%H A366333 Eduard I. Vatutin, <a href="http://evatutin.narod.ru/spectra/spectra_cyclic_dls_diagonal_transversals_all.png">Graphical representation of the spectra</a>.
%H A366333 <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A366333 For n=6*2+1=13 the number of diagonal transversals that a semicyclic diagonal Latin square of order 13 may have is 127339, 127830, 128489, 128519, 128533, 128608, 128751, 128818, 128861, 129046, 129059, 129171, 129243, 129286, 129353, 129474, 129641, 129657, 130323 or 131106. Since there are 20 distinct values, a(6)=20.
%Y A366333 Cf. A071607, A341585, A342990, A345370, A366331.
%K A366333 nonn,more,hard
%O A366333 0,6
%A A366333 _Eduard I. Vatutin_, Oct 07 2023