A368499 -id:A368499 - cp's OEIS frontend

A366778 Number of nonequivalent cycles of length 2n in the (2n+1) X (2n+1) knight graph.

3, 25, 480, 12000, 350256, 10780549, 344680960

Offset: 2

Stoyan Kapralov, Dec 15 2023

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.

			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.

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.

Cf. A368499, A001230, A234623, A254129, A356404.

cp's OEIS Frontend

A366778 Number of nonequivalent cycles of length 2n in the (2n+1) X (2n+1) knight graph.

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