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> "Let's define $a\_n=\sum\limits\_{k=0}^{\lfloor n/2 \rfloor} {n-k \choose k}\left(-\frac{1}{4}\right)^k$. Evaluate $a\_{1997}$."  
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This problem is from the final round of an old South Korean Mathematical Olympiad (1997 KMO).   

I think this problem is very simple, but requires some combinatoric ideas, and also is very interesting.  

But as a lot of time has passed by, I cannot find any solutions or guidelines about it.   

I tried to divide the explicit form of $(x+y)^{2k}$ with $x^k$, but it doesn't work well.   

Would you help me?