# Problem 400: Fibonacci tree game
A Fibonacci tree is a binary tree recursively defined as:T(0) is the
empty tree. T(1) is the binary tree with only one node. T(k) consists of
a root node that has T(k-1) and T(k-2) as children. On such a tree two
players play a take-away game. On each turn a player selects a node and
removes that node along with the subtree rooted at that node. The player
who is forced to take the root node of the entire tree loses. Here are
the winning moves of the first player on the first turn for T(k) from
k=1 to k=6. Let f(k) be the number of winning moves of the first player
(i.e. the moves for which the second player has no winning strategy) on
the first turn of the game when this game is played on T(k). For
example, f(5) = 1 and f(10) = 17. Find f(10000). Give the last 18 digits
of your answer.