Bob and Alice are playing a game. They will start with an integer $n$. Alice goes first, in each turn, a player can choose an integer between 1 and 13 and that number is to be subtracted from $n$. They will repeat this process alternatively. The game ends when $n$ becomes less than 1. The person who will be the telling the last number will lose the game.
Given $n$ (initial value), how could we determine the value of $n$ after the $k$th turn of Alice (If Alice plays optimally)?
PS: For this particular puzzle, it is given $n = 1251$ and $k = 19$. However, I am interested in the general solution.