See A079137, which is the main entry for this problem.
This sequence is known to be given by a linear recurrence relation with constant coefficients, although as far as I know this recurrence has not yet been explicitly computed.
Table of n, a(n) for n=1..21.
There are 2976 ways to start with a knight on some square of a 4 X 6 chessboard and make 23 moves such that each square is visited exactly once.
Equals 4*A079137(n). Cf. A070030.
Sequence in context: A023927 A240253 A057376 * A221423 A062543 A086820
Adjacent sequences: A079309 A079310 A079311 * A079313 A079314 A079315
Alex Healy, Feb 11 2003