%I #18 Dec 12 2023 20:45:34
%S 1,0,1,1,0,2,20,0,272,1208,0,127334,1958084,0
%N Number of main classes of diagonal Latin squares of order 2n+1 that contain a horizontally semicyclic Latin square.
%C 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).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1911">About the horizontally and vertically semicyclic diagonal Latin squares enumeration</a> (in Russian).
%H 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 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2450">About the number of main classes of semicyclic diagonal Latin squares of order 17</a> (in Russian).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2453">About the number of main classes of semicyclic diagonal Latin squares of order 19</a> (in Russian).
%H Eduard I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_horz_semicyclic_cfs_n5-19.zip">Lists of canonical forms of semicyclic diagonal Latin squares of orders 5-19</a>.
%H <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e Example of horizontally semicyclic diagonal Latin square of order 13:
%e 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 2 3 4 5 6 7 8 9 10 11 12 0 1 (d=2)
%e 4 5 6 7 8 9 10 11 12 0 1 2 3 (d=4)
%e 9 10 11 12 0 1 2 3 4 5 6 7 8 (d=9)
%e 7 8 9 10 11 12 0 1 2 3 4 5 6 (d=7)
%e 12 0 1 2 3 4 5 6 7 8 9 10 11 (d=12)
%e 3 4 5 6 7 8 9 10 11 12 0 1 2 (d=3)
%e 11 12 0 1 2 3 4 5 6 7 8 9 10 (d=11)
%e 6 7 8 9 10 11 12 0 1 2 3 4 5 (d=6)
%e 1 2 3 4 5 6 7 8 9 10 11 12 0 (d=1)
%e 5 6 7 8 9 10 11 12 0 1 2 3 4 (d=5)
%e 10 11 12 0 1 2 3 4 5 6 7 8 9 (d=10)
%e 8 9 10 11 12 0 1 2 3 4 5 6 7 (d=8)
%Y Cf. A071607, A123565, A287764, A338562, A342990, A343866.
%K nonn,more,hard
%O 0,6
%A _Eduard I. Vatutin_, Oct 07 2023
%E a(11)-a(13) from _Andrew Howroyd_, Nov 02 2023