I don't even...

Define a subset SS of the first 3030 positive integers to be uneven if, for all iSi \in S, i+2Si+2 \notin S. For example, {1,2}\{1, 2\} is an uneven subset, while {1,2,3}\{1, 2, 3\} is not. If NN represents the number of uneven subsets, find the remainder when NN is divided by 10001000.

Notes:

The empty set is considered to be an uneven subset.

You may want use a calculator at the end.

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