I don't even...

Define a subset \(S\) of the first \(30\) positive integers to be uneven if, for all \(i \in S\), \(i+2 \notin S\). For example, \(\{1, 2\}\) is an uneven subset, while \(\{1, 2, 3\}\) is not. If \(N\) represents the number of uneven subsets, find the remainder when \(N\) is divided by \(1000\).

Notes:

The empty set is considered to be an uneven subset.

You may want use a calculator at the end.

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