cp's OEIS Frontend

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+2k)-p(i)|<>6k AND |p(j+6k)-p(j)|<>2k for all i>=1, j>=1, k>=1, i+2k<=n, j+6k<=n.

a(n) is the number of ways to pair the natural numbers from 1 to n with those between n+1 and 2*n into n pairs (xi,yi) such that the 2*n numbers yi+i and yi-i are all different. - Michel Marcus, Apr 27 2016

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.

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.

The term a(4) with the value 26 is the count for a board size of 4 squares by 4 squares. The highest term so far a(45) is the count for a board of 45 squares by 45 squares.

This whole sequence refers only to the number of queen pieces placed tentatively on a board in the hunt for the FIRST POSSIBLE solution for each board size. This sequence makes no reference to queen placements needed to hunt for subsequent solutions that are possible for board sizes above 3x3.

In other words, number of maximal independent vertex sets (and minimal vertex covers) in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017

a(n) is also number of permutations p of 1,2,...,n satisfying |p(j+5k)-p(j)|<>5k for all j>=1, k>=1, j+5k<=n

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+3k)-p(i)|<>5k AND |p(j+5k)-p(j)|<>3k for all i>=1, j>=1, k>=1, i+3k<=n, j+5k<=n

a(n) is also number of permutations p of 1,2,...,n satisfying |p(i+3k)-p(i)|<>6k AND |p(j+6k)-p(j)|<>3k for all i>=1, j>=1, k>=1, i+3k<=n, j+6k<=n