A polyknight is a plane geometric figure formed by selecting cells in a square lattice that could represent the path of a chess knight in which doubling back is allowed. It is a polyform with square cells which are not necessarily connected, comparable to the polyking. Alternatively, it can be interpreted as a connected subset of the vertices of a knight's graph, a graph formed by connecting pairs of lattice squares that are a knight's move apart..
Enumeration of polyknights Free, one-sided, and fixed polyknights Three common ways of distinguishing polyominoes for enumeration can also be extended to polyknights: free polyknights are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another (pieces that can be picked up and flipped over). one-sided polyknights are distinct when none is a translation or rotation of another (pieces that cannot be flipped over). fixed polyknights are distinct when none is a translation of another (pieces that can be neither flipped nor rotated).
The following table shows the numbers of polyknights of various types with n cells. n free one-sided fixed 1 1 1 1 2 1 2 4 3 6 8 28 4 35 68 234 5 290 550 2,162 6 2,680 5,328 20,972 7 26,379 52,484 209,608 8 267,598 534,793 2,135,572 9 2,758,016 5,513,338 22,049,959 10 28,749,456 57,494,308 229,939,414 OEIS
Notes