cp's OEIS Frontend

This sequence counts classes of "near n-queens solutions". Permutations with at most 1 queen on any torus diagonal are exactly the torus n queen solutions (A007705), those with at most 2 contain the normal n queen solutions (A000170).

Therefore they may be called "near n-queens solutions". In this sequence, permutations p and q are considered equivalent iff there are natural 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, rotations, reflections and torus shifts are allowed. The sequence contains the objects of A062164.

Two n-queens solutions 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 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 reduces exactly the objects of A062164 and, via that sequence, these of A002562 and A000170. Note that the equivalence classes of this sequence are a subset of A062168.