OFFSET
2,1
COMMENTS
A knight graph is a graph that represents all legal moves of the knight chess piece on a chessboard. Each vertex of this graph represents a square of the chessboard, and each edge connects two squares that are a knight's move apart from each other.
Two cycles in the knight graph are called equivalent if one can be obtained from another by applying one or more of the operations of translation, rotation, and symmetry on the chessboard; otherwise, they are nonequivalent.
LINKS
Stoyan Kapralov, Valentin Bakoev, and Kaloyan Kapralov, Algorithms for Construction and Enumeration of Closed Knight's Paths, Mathematics and Informatics, 2, (2023), 107-114; arXiv:2304.00565 [math.CO], 2023.
EXAMPLE
For n=2 the a(2)=3 solutions (in standard chess notation) are: (a1, c2, d4, b3), (a2, c1, d2, c3), and (a2, c1, d3, b3).
Note that each of these three cycles is non-self-intersecting. For the remaining values of n there are two kind of cycles - self-intersecting and non-self-intersecting. For example, a self-intersecting cycle of length 6 is (a1, c2, b4, a2, c1, b3), while the cycle (a1, c2, e1, f3, d4, b3) is non-self-intersecting.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Stoyan Kapralov, Dec 15 2023
STATUS
approved