Problem - Stone Game IV

Alice and Bob play with $n$ stones. Each turn, remove a positive square number of stones ($1, 4, 9, 16, ...$). The player who cannot move loses. Determine if Alice (first player) wins with optimal play.

Example: $n = 7$. Alice takes $4$, leaving $3$. Bob takes $1$, leaving $2$. Alice takes $1$, leaving $1$. Bob takes $1$, leaving $0$. Alice can't move. Bob wins.