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 A372922 #29 Jan 11 2025 04:12:37 %S A372922 1,0,480,161280,2229534720,45984153600000,3271798279766016000 %N A372922 Number of diagonal Latin squares of order 2n+1 that are isomorphic to cyclic Latin squares by row and column permutations. %C A372922 The Latin squares considered here are diagonal Latin squares that are isomorphic to cyclic Latin squares. They are can be obtained from cyclic Latin squares (see A338522) by diagonalization (getting a corresponding pair of transversals and placing them on the diagonals, see article). These Latin squares have some interesting properties, for example, there are a large number of diagonal transversals. %H A372922 Eduard 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 A372922 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2443">About the different types of cyclic diagonal Latin squares</a> (in Russian). %H A372922 E. Vatutin, A. Belyshev, N. Nikitina, M. Manzuk, A. Albertian, I. Kurochkin, A. Kripachev, and A. Pykhtin, <a href="https://doi.org/10.1007/978-3-031-49435-2_4">Diagonalization and Canonization of Latin Squares</a>, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023. pp. 48-61. %H A372922 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>. %F A372922 a(n) = A372923(n) * (2n+1)!. - _Eduard I. Vatutin_, Sep 08 2024 %e A372922 The cyclic Latin square of order 7 %e A372922 . %e A372922 0 1 2 3 4 5 6 %e A372922 1 2 3 4 5 6 0 %e A372922 2 3 4 5 6 0 1 %e A372922 3 4 5 6 0 1 2 %e A372922 4 5 6 0 1 2 3 %e A372922 5 6 0 1 2 3 4 %e A372922 6 0 1 2 3 4 5 %e A372922 . %e A372922 has a pair of symmetrically placed transversals T1 = (0, 2, 4, 6, 1, 3, 5) and T2 = (0, 5, 3, 1, 6, 4, 2), after permutting rown and columns transversal T1 placed to the main diagonal with getting single diagonal Latin square %e A372922 . %e A372922 2 5 0 3 4 6 1 %e A372922 0 3 5 1 2 4 6 %e A372922 1 4 6 2 3 5 0 %e A372922 6 2 4 0 1 3 5 %e A372922 3 6 1 4 5 0 2 %e A372922 4 0 2 5 6 1 3 %e A372922 5 1 3 6 0 2 4 %e A372922 . %e A372922 then after permuting rows and columns transversal T2 placed to the second diagonal with getting diagonal Latin square %e A372922 . %e A372922 2 5 0 3 6 1 4 %e A372922 0 3 5 1 4 6 2 %e A372922 1 4 6 2 5 0 3 %e A372922 6 2 4 0 3 5 1 %e A372922 4 0 2 5 1 3 6 %e A372922 5 1 3 6 2 4 0 %e A372922 3 6 1 4 0 2 5 %e A372922 . %e A372922 that can be canonized to the following diagonal Latin square: %e A372922 . %e A372922 0 1 2 3 4 5 6 %e A372922 2 3 1 5 6 4 0 %e A372922 5 6 4 0 1 2 3 %e A372922 4 0 6 2 3 1 5 %e A372922 6 2 0 1 5 3 4 %e A372922 1 5 3 4 0 6 2 %e A372922 3 4 5 6 2 0 1 %e A372922 . %e A372922 Cyclic Latin square of order 11 %e A372922 . %e A372922 0 1 2 3 4 5 6 7 8 9 10 %e A372922 1 2 3 4 5 6 7 8 9 10 0 %e A372922 2 3 4 5 6 7 8 9 10 0 1 %e A372922 3 4 5 6 7 8 9 10 0 1 2 %e A372922 4 5 6 7 8 9 10 0 1 2 3 %e A372922 5 6 7 8 9 10 0 1 2 3 4 %e A372922 6 7 8 9 10 0 1 2 3 4 5 %e A372922 7 8 9 10 0 1 2 3 4 5 6 %e A372922 8 9 10 0 1 2 3 4 5 6 7 %e A372922 9 10 0 1 2 3 4 5 6 7 8 %e A372922 10 0 1 2 3 4 5 6 7 8 9 %e A372922 . %e A372922 can be diagonalized to set of diagonal Latin squares: %e A372922 . %e A372922 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 %e A372922 1 2 3 4 5 10 8 9 0 6 7 1 2 3 4 6 7 8 9 10 0 5 1 2 3 4 5 10 9 0 7 8 6 %e A372922 3 4 5 10 7 9 1 8 2 0 6 8 10 5 7 9 3 0 4 1 6 2 3 4 5 10 6 9 7 2 1 0 8 %e A372922 4 5 10 7 9 6 2 0 3 1 8 4 6 8 10 5 1 7 2 9 3 0 10 6 9 8 7 0 2 5 4 3 1 %e A372922 10 7 9 6 8 0 4 2 5 3 1 9 0 1 2 3 10 4 5 6 7 8 9 8 7 0 1 2 4 6 10 5 3 %e A372922 7 9 6 8 0 1 5 3 10 4 2 7 9 0 1 2 8 3 10 4 5 6 5 10 6 9 8 7 1 4 3 2 0 %e A372922 8 0 1 2 3 4 9 10 6 7 5 6 8 10 5 7 2 9 3 0 4 1 7 0 1 2 3 4 10 8 9 6 5 %e A372922 2 3 4 5 10 7 0 6 1 8 9 10 5 7 9 0 4 1 6 2 8 3 4 5 10 6 9 8 0 3 2 1 7 %e A372922 5 10 7 9 6 8 3 1 4 2 0 3 4 6 8 10 0 5 1 7 2 9 8 7 0 1 2 3 5 9 6 10 4 %e A372922 6 8 0 1 2 3 7 5 9 10 4 2 3 4 6 8 9 10 0 5 1 7 6 9 8 7 0 1 3 10 5 4 2 %e A372922 9 6 8 0 1 2 10 4 7 5 3 5 7 9 0 1 6 2 8 3 10 4 2 3 4 5 10 6 8 1 0 7 9 ... %e A372922 . %e A372922 (totally 81 main classes of diagonal Latin squares). %Y A372922 Cf. A338522, A338562, A342990, A372923, A375475. %K A372922 nonn,more,hard %O A372922 0,3 %A A372922 _Eduard I. Vatutin_, May 16 2024