Polyknight
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.^{[1]}
Enumeration of polyknights
Free, onesided, and fixed polyknights
Three common ways of distinguishing polyominoes for enumeration^{[2]} 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).
 onesided 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  onesided  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  A030446  A030445  A030444 
Free polyknights  


Notes
 ^ Aleksandrowicz, Gadi; Barequet, Gill (2011), "Parallel enumeration of lattice animals", in Atallah, Mikhail J.; Li, XiangYang; Zhu, Binhai, Frontiers in Algorithmics and Algorithmic Aspects in Information and Management  Joint International Conference, FAWAAIM 2011, Jinhua, China, May 2831, 2011. Proceedings, Lecture Notes in Computer Science, 6681, Springer, pp. 90–99, doi:10.1007/9783642212048_13.
 ^ Redelmeier, D. Hugh (1981), "Counting polyominoes: yet another attack", Discrete Mathematics, 36: 191–203, doi:10.1016/0012365X(81)902375
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