Willy's New Game
Willy has just created a new game. In this game, two players play against each other in a desperate battle to count to the number \(P\). Each turn, the player must count one, two, or three consecutive natural numbers, and must start where the previous player stopped counting on the previous turn. The first player must start counting at \(1\).
Sample Game (with \(P = 7\)):
Player 1 says, "\(1, 2, 3\)".
Player 2 says, "\(4\)".
Player 1 says, "\(5, 6\)".
Player 2 says, "\(7\)".
The game stops after Player 2 says \(7\) because \(P = 7\). Note that the length of each turn is the amount of numbers said during the turn (for example, the first turn in the above game has length 3).
How many different games are possible if \(P = 100\)?
Edit: Note that a game is different than another game if the number of turns of length one are different for both games, if the number of turns of length two are different for both games, or if the number of turns of length three are different for both games.