# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a341585 Showing 1-1 of 1 %I A341585 #46 Mar 22 2022 15:33:50 %S A341585 1,0,1,1,0,2,3,0,4,4,0,5,1,0,7,7,0,1,9,0,10,10,0,11,1,0,13,2,0,14,15, %T A341585 0,3,16,0,17,18,0,4,19,0,20,4,0,22,5,0,4,24,0,25,25,0,26,27,0,28,5,0, %U A341585 7,2,0,1,31,0,32,8,0,34,34,0,10,7,0,37,37,0,7,39,0,10 %N A341585 Number of main classes of cyclic diagonal Latin squares of order 2n+1. %C A341585 There are no cyclic diagonal Latin squares of even order. %C A341585 All cyclic diagonal Latin squares are pandiagonal. Conversely, all pandiagonal Latin squares are cyclic for orders 5, 7 and 11. %C A341585 From _Andrew Howroyd_, May 01 2021: (Start) %C A341585 Depending on exactly which Latin squares constitute a main class, slightly different sequences are possible. Another variation is given in A343866. %C A341585 In this sequence equivalence allows for the permutation of symbols, the transpose of rows with columns, and any permutation of rows and columns that preserves the cyclic and diagonal properties. This permutation must transform every cyclic diagonal Latin square into another, but does not necessarily transform an arbitrary diagonal Latin square that is not cyclic into another diagonal Latin square. %C A341585 The row (or column) permutations that satisfy this requirement form a group and are those where every k-th row (or column) is taken cyclically where k is any number that is congruent to 1 or -1 modulo every prime divisor of the order of the Latin square. (End) %H A341585 Eduard I. Vatutin, About the number of main classes of pandiagonal Latin squares of orders 1-12 and cyclic diagonal Latin squares of orders 1-22 (in Russian). %H A341585 Eduard I. Vatutin, About the number of cyclic diagonal Latin squares of orders 23-24 (in Russian). %H A341585 Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition — 2021, pp. 77-79. (in Russian) %H A341585 Eduard I. Vatutin, Proving list. %H A341585 Index entries for sequences related to Latin squares and rectangles. %F A341585 a((p-1)/2) = A343866((p-1)/2) for odd prime p. - _Andrew Howroyd_, May 02 2021 %e A341585 For n=0 there is only 1 Latin square of order 1, so a(0)=1. %e A341585 For n=2 there is one main class with canonical form (CF) of cyclic diagonal Latin squares of order 2n+1=5: %e A341585 0 1 2 3 4 %e A341585 2 3 4 0 1 %e A341585 4 0 1 2 3 %e A341585 1 2 3 4 0 %e A341585 3 4 0 1 2 %e A341585 so a(2)=1. %e A341585 For n=3 there is one main class of order 7 with CF: %e A341585 0 1 2 3 4 5 6 %e A341585 2 3 4 5 6 0 1 %e A341585 4 5 6 0 1 2 3 %e A341585 6 0 1 2 3 4 5 %e A341585 1 2 3 4 5 6 0 %e A341585 3 4 5 6 0 1 2 %e A341585 5 6 0 1 2 3 4 %e A341585 so a(3)=1. %e A341585 a(12) = 1. There are A123565(25) = 10 cyclic diagonal Latin squares whose first row is in ascending order. The 10 row permutations constructed by selecting every k-th row cyclically where k is one of 1, 4, 6, 9, 11, 14, 16, 19, 21, 24 (numbers congruent to 1 or -1 modulo 5) transforms each of these between each other so there is only a single class. - _Andrew Howroyd_, May 02 2021 %o A341585 (PARI) %o A341585 G(n)={my(f=factor(n)[,1]); select((d->for(i=1, #f, if((d-1)%f[i]&&(d+1)%f[i], return(0)));1), [1..n])} %o A341585 iscanon(n,k,g) = k <= vecmin(g*k%n) && k <= vecmin(g*lift(1/Mod(k,n))%n) %o A341585 a(n)={if(n==0, 1, my(m=2*n+1, g=G(m)); sum(k=1, m-1, gcd(m,k)==1 && gcd(m,k-1)==1 && gcd(m,k+1)==1 && iscanon(m, k, g)))} \\ _Andrew Howroyd_, Apr 30 2021 %Y A341585 Cf. A123565, A338562, A343866. %K A341585 nonn %O A341585 0,6 %A A341585 _Eduard I. Vatutin_, Feb 15 2021 %E A341585 Offset corrected and terms a(12) and beyond from _Andrew Howroyd_, Apr 30 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE