Problem 45389. Knight's Watch

"Night gathers, and now my watch begins"

A knight is placed on an n-by-n sized chessboard at the position x. Find the probability that after k steps, the knight will remain within the chessboard.

Any knight's move that places him outside the board should be considered invalid.

 For simplicity, the knight's position on the chessboard is defined with the numeric
 notation instead of algebraic notation. so 'Ka1' is represented as (1,1).

Brief explanation:

Say the knight is placed in pos-(1,1). A knight has 8 possible moves. So in the next move, 
the Knight can go to 8 different positions in the chessboard. But among them, only 2
positions are valid i.e. the knight remains within the chessboard and they are -
(3,2) & (2,3). So the prob. is 2/8 after 1 move. What will be the probability after k moves?