Two players play a game in which they alternate calling out positive integers ≤ N, according to:

The first player must always call out odd numbers.

The second player must always call out even numbers.

Each player must call out a number greater than the previously called number (except, obviously, the very first time).

The player who cannot call out a number loses.

How many different possible games are there? And, if we count a turn each time a player calls out a number, how many different K-turns games are there?

Note: the game is not very fun to play (why?) but the puzzles are interesting!