A343866 Number of inequivalent cyclic diagonal Latin squares of order 2n+1 up to rotations, reflections and permutation of symbols.

1, 0, 1, 1, 0, 2, 3, 0, 4, 4, 0, 5, 3, 0, 7, 7, 0, 2, 9, 0, 10, 10, 0, 11, 7, 0, 13, 4, 0, 14, 15, 0, 6, 16, 0, 17, 18, 0, 8, 19, 0, 20, 8, 0, 22, 10, 0, 8, 24, 0, 25, 25, 0, 26, 27, 0, 28, 10, 0, 14, 22, 0, 13, 31, 0, 32, 16, 0, 34, 34, 0, 20, 14, 0, 37, 37, 0, 14, 39, 0, 20

Offset

0,6

Comments

Also the number of main classes of diagonal Latin squares of order 2n+1 that contain a cyclic Latin square. Compare A341585.

Links

Table of n, a(n) for n=0..80.

Formula

a((p-1)/2) = A341585((p-1)/2) for odd prime p.

Example

a(12) = 3 since there are A123565(25) = 10 cyclic diagonal Latin squares whose first row is in ascending order. Each of these is uniquely defined by the step between rows and form 5 pairs by horizontal or vertical reflection (negating the step between rows). Up to exchanging rows with columns there are 3 distinct classes, so a(12) = 3.

Prog

(PARI)
iscanon(n, k, g) = k <= vecmin(g*k%n) && k <= vecmin(g*lift(1/Mod(k, n))%n)
a(n)={if(n==0, 1, my(m=2*n+1); sum(k=1, m-1, gcd(m, k)==1 && gcd(m, k-1)==1 && gcd(m, k+1)==1 && iscanon(m, k, [1, -1])))}

Crossrefs

Cf. A123565, A287764, A338562, A341585.

Sequence in context: A230451 A286239 A341585 * A140502 A175434 A154860

Adjacent sequences: A343863 A343864 A343865 * A343867 A343868 A343869

Keyword

nonn

Author

Andrew Howroyd, May 02 2021

Status

approved