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This question is related to a few questions which have been posted on the website :

Because of the irrationality of $\pi$, the sequence $(\cos(n))_{n\in\mathbb{N}}$ is dense in $[-1;1]$. For any value $a\in[-1;1]$, we can extract a subsequence $(\cos(n_k))_{k\in\mathbb{N}}$ convergent to $a$.

My question is the following: Does someone know an example of a convergent subsequence of $\cos(n)$ with an explicit expression?

Some more comments:

We could define a subsequence in the following way: $$n_0=1;\quad n_{k+1}>n_k \text{ such that } |\cos(n_k)-a|<1/k.$$ This subsequence is well defined (and unique if we add the condition that $n_{k+1}$ should be minimum) and converges. But I would say, that this not explicit. I don't have a definition of what should be an explicit expression, and any answer are welcome.

Reading the nice answer of David Speyer in Is there a limit of $\cos(n!)$, it seems that we still don't understand enough about $\pi$ to proof or disprove that $\cos(n!)$ diverges. Because of these comments, I would not be surprised if the answer to my question is no.

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Gilles Bonnet
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    Does "$n_k$ is the numerator of the $k^{\rm th}$ convergent in the continued fraction expansion of $2\pi$" count as an explicit expression? – Micah Feb 17 '14 at 20:56
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    @Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2\pi$"? ;-) I am looking for something more "explicit". – Gilles Bonnet Feb 17 '14 at 21:01

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