# 2010-21 Limit

Let $$a_1=0$$, $$a_{2n+1}=a_{2n}=n-a_n$$. Prove that there exists k such that $$\lvert a_k- \frac{k}{3}\rvert >2010$$ and yet $$\lim_{n\to \infty} \frac{a_n}{n}=\frac13$$.

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