Double Recursion

Consider the two recursive relations.

$p_0 = 0, p_1 = 1, p_n = 2p_{n-1} + p_{n-2}, n \geq 2$

$q_0 = 1, p_1 = 1, q_n = 2q_{n-1} + q_{n-2}, n \geq 2$

If $\arctan \left ( \sqrt[2013]{\sqrt{2} p_{2013} + q_{2013} } \right ) = \frac {a }{b} \pi$ where $a$ and $b$ are coprime positive integers. What is the value of $a +b$ ?

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$\Large \color{#3D99F6}{\text{Hi Calvin! Read here onwards!} }$

During a lunch break, there are 15 workers sitting around a rotatable round table such that they are all equally spaced from their neighboring colleagues. They decided to open the letters the mailman delivered to them. However, due to the mailman's negligence, everyone's mail got mixed up and none of them received the right mail. These workers decided to rotate the table such that at least one of them are sitting in front of their mail.

With all possible rotations, what is minimum amount of people that are sitting in front of letter that is intended for them?

Logic > Miscellaneous Grid Puzzles > LVL 4 (bordering LVL5 I believe)

Answer choices
1
2 <<< Correct
3
4
5

Notes:

1. Source

2. There is no particular reason why I chose 15. It could be any integer above 2. I just don't want to use the general $n$ , which makes it more math oriented.

3. Might be a good idea to use "Brilliant.org office meeting" as the setting of this problem.