The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f1,f2,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size fi. It is proved that there exist constants c=c(n) and k0=k0(n) such that fk+1=fk+fk−c for all k>k0, and limn→∞ c(n)/(nlogn)=1.