A man walks $n$ blocks north and $3n$ blocks east from his residence at $(0,0)$. How many routes are possible from his home to his office at $(3n, n)$ if he cannot walk two block north consecutively?
It is obvious that the good paths are $\binom{3n}{n}$ but how to calculate the bad paths in this walk which would be any two consecutive blocks north?
Almost every north must be followed by an east, so make the moves east and northeast. The only exception is if the path ends with a north, so count those separately.