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 A342990 #25 Apr 04 2024 10:46:09 %S A342990 1,0,240,20160,0,319334400,2167003238400,0,2943669154922496000, %T A342990 5253122016055001088000,0,144827547726179682893168640000, %U A342990 1214667347283206181421056000000000,0,184737047979495031539522261089255424000000,3555700708206908663181998415125686517760000000,0 %N A342990 Number of horizontally or vertically semicyclic diagonal Latin squares of order 2n+1. %C A342990 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). 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. Definition from A343867 includes this type of squares but not only it. %H A342990 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1911">About the horizontally and vertically semicyclic diagonal Latin squares enumeration</a> (in Russian). %H A342990 E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_dls_spec_types_list.pdf">Special types of diagonal Latin squares</a>, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian) %H A342990 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2691">Numerical formula between number of horizontally or vertically semicyclic diagonal Latin squares and number of toroidal n-queens problem solutions</a> (in Russian). %H A342990 <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>. %F A342990 a(n) = A071607(n) * (2*n+1)!. %F A342990 a(n) = A007705(n) * (2n)!. - _Eduard I. Vatutin_, Mar 15 2024 %e A342990 Example of cyclic diagonal Latin square of order 13: %e A342990 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A342990 2 3 4 5 6 7 8 9 10 11 12 0 1 (d=2) %e A342990 4 5 6 7 8 9 10 11 12 0 1 2 3 (d=4) %e A342990 6 7 8 9 10 11 12 0 1 2 3 4 5 (d=6) %e A342990 8 9 10 11 12 0 1 2 3 4 5 6 7 (d=8) %e A342990 10 11 12 0 1 2 3 4 5 6 7 8 9 (d=10) %e A342990 12 0 1 2 3 4 5 6 7 8 9 10 11 (d=12) %e A342990 1 2 3 4 5 6 7 8 9 10 11 12 0 (d=14 == 1 (mod 13)) %e A342990 3 4 5 6 7 8 9 10 11 12 0 1 2 (d=16 == 3 (mod 13)) %e A342990 5 6 7 8 9 10 11 12 0 1 2 3 4 (d=18 == 5 (mod 13)) %e A342990 7 8 9 10 11 12 0 1 2 3 4 5 6 (d=20 == 7 (mod 13)) %e A342990 9 10 11 12 0 1 2 3 4 5 6 7 8 (d=22 == 9 (mod 13)) %e A342990 11 12 0 1 2 3 4 5 6 7 8 9 10 (d=24 == 11 (mod 13)) %e A342990 Example of horizontally semicyclic diagonal Latin square of order 13: %e A342990 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A342990 2 3 4 5 6 7 8 9 10 11 12 0 1 (d=2) %e A342990 4 5 6 7 8 9 10 11 12 0 1 2 3 (d=4) %e A342990 9 10 11 12 0 1 2 3 4 5 6 7 8 (d=9) %e A342990 7 8 9 10 11 12 0 1 2 3 4 5 6 (d=7) %e A342990 12 0 1 2 3 4 5 6 7 8 9 10 11 (d=12) %e A342990 3 4 5 6 7 8 9 10 11 12 0 1 2 (d=3) %e A342990 11 12 0 1 2 3 4 5 6 7 8 9 10 (d=11) %e A342990 6 7 8 9 10 11 12 0 1 2 3 4 5 (d=6) %e A342990 1 2 3 4 5 6 7 8 9 10 11 12 0 (d=1) %e A342990 5 6 7 8 9 10 11 12 0 1 2 3 4 (d=5) %e A342990 10 11 12 0 1 2 3 4 5 6 7 8 9 (d=10) %e A342990 8 9 10 11 12 0 1 2 3 4 5 6 7 (d=8) %Y A342990 Cf. A007705, A071607, A338562, A343867. %K A342990 nonn,more,hard %O A342990 0,3 %A A342990 _Eduard I. Vatutin_, Jan 27 2022