Well, it's a matter of terminology - I never talked to your teacher. In my notations you count this way

m=2:

1 2 (out) 3 4 (out) ...

m=3:

1 2 3 (out) ...

My m is 1 more than yours.

I am pretty sure no one has a recursive solution for m = 2 in your notations. You may try another solution on my page that does not produce a formula but leads to an answer.

Regards,
Alexander Bogomolny