This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343867 #43 May 18 2024 14:52:08 %S A343867 0,0,0,0,0,0,1560,0,34000,175104,0,22417824,313235960,0,83574857328, %T A343867 1729671003296 %N A343867 Number of semicyclic pandiagonal Latin squares of order 2*n+1 with the first row in ascending order. %C A343867 Pandiagonal Latin squares exist only for odd orders not divisible by 3. All pandiagonal Latin squares for orders less than 13 are cyclic which are not counted by this sequence. %C A343867 Semicyclic Latin squares are defined in the Atkin reference where the first nonzero term of this sequence is given. They are cyclic in a single direction. The direction can be horizontal or vertical or any other step such as a knights move. %C A343867 Each symbol in a semicyclic Latin square occupies the same pattern of squares up to translation on the torus which in the case of a pandiagonal square is a solution to the toroidal n-queens problem. %C A343867 For prime 2n+1, a(n) is a multiple of 2n+1. %H A343867 A. O. L. Atkin, L. Hay, and R. G. Larson, <a href="https://doi.org/10.1016/0898-1221(83)90130-X">Enumeration and construction of pandiagonal Latin squares of prime order</a>, Computers & Mathematics with Applications, Volume. 9, Iss. 2, 1983, pp. 267-292. %H A343867 Andrew Howroyd, <a href="/A343867/a343867.txt">PARI Program for Initial Terms</a>. %H A343867 Natalia Makarova from Harry White, <a href="https://disk.yandex.ru/d/3KpNZgnH19a0Vg">1560 semi-cyclic Latin squares of order 13</a>. %H A343867 Natalia Makarova from Harry White, <a href="https://disk.yandex.ru/d/myvKxPwu4q3gAg">34000 semi-cyclic Latin squares of order 17</a>. %H A343867 Eduard I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_ahl_pandiagonal_n19_175104_items_without_cyclic.zip">175104 semi-cyclic Latin squares of order 19</a>. %F A343867 a(n) >= 4*(A071607(n) - A123565(2*n+1)). %e A343867 The following is an example of an order 13 semicyclic square with a step of (1,4). This means moving down one row and across by 4 columns increases the cell value by 1 modulo 13. Symbols can be relabeled to give a square with the first row in ascending order. %e A343867 0 11 1 7 5 9 3 10 4 8 6 12 2 %e A343867 9 7 0 3 1 12 2 8 6 10 4 11 5 %e A343867 11 5 12 6 10 8 1 4 2 0 3 9 7 %e A343867 1 4 10 8 12 6 0 7 11 9 2 5 3 %e A343867 10 3 6 4 2 5 11 9 0 7 1 8 12 %e A343867 8 2 9 0 11 4 7 5 3 6 12 10 1 %e A343867 7 0 11 2 9 3 10 1 12 5 8 6 4 %e A343867 6 9 7 5 8 1 12 3 10 4 11 2 0 %e A343867 5 12 3 1 7 10 8 6 9 2 0 4 11 %e A343867 3 1 5 12 6 0 4 2 8 11 9 7 10 %e A343867 12 10 8 11 4 2 6 0 7 1 5 3 9 %e A343867 2 6 4 10 0 11 9 12 5 3 7 1 8 %e A343867 4 8 2 9 3 7 5 11 1 12 10 0 6 %e A343867 ... %e A343867 a(12) = 4*(A071607(12) - A123565(25)) + 11240. - _Jim White_, Jul 22 2021 %e A343867 a(14) = 4*(A071607(14) - A123565(29)) + 91176. - _Jim White_, Jul 24 2021 %e A343867 a(15) = 4*(A071607(15) - A123565(31)) + 334800. - _Jim White_, Aug 03 2021 %o A343867 (PARI) \\ See Links %Y A343867 Cf. A071607, A123565 (cyclic), A338620, A343868. %K A343867 nonn,more %O A343867 0,7 %A A343867 _Andrew Howroyd_, May 08 2021 %E A343867 a(12)-a(15) from _Jim White_, Aug 03 2021