cp's OEIS Frontend

In this sequence two n-queens solutions p and q are considered equivalent iff there are natural numbers x and y such that, for all k from {0, ..., n-1}, q (k + x mod n) = p (k) + y mod n, or q is a rotation or a reflection of such a q.

In other words, besides rotations and reflections, also torus shifts are allowed. The sequence reduces the objects of A002562 and via that of A000170. The reduction of A000170 to this sequence is exactly the same as from A007705 to A053994 for torus queens; however, a solution for torus queens remains always a solution after a shift while a normal queens solutions does so only sometimes.

Note that the equivalence classes of this sequence are a subset of A006841. Moreover they are a subset of A062167.

As A062167, also this sequence counts classes of "near n-queens solutions". In this sequence, two permutations p and q are considered similar iff there is a factor f, 0 < f < n, satisfying GCD (f,n) = 1, such that for all k from {0, ..., n-1} q (k * f mod n) = p (k) * f mod n or if q is a rotation, a reflection or a shift of such a q. In other words, also expansions are allowed which move the queen at (k, p(k)) to (f * k mod n, f * p(k) mod n). The sequence contains the objects of A062165.