This question has deceptively simple constraints.
The pattern seems quite unusual, but it is not difficult to see a recursive definition for G(n) that behaves similarly to the Fibonacci sequence, with a twist. showing that what I have is indeed the recursive formula is probably is sufficient for the proof part of the question
The hard part of the question is in proving that my formula is indeed correct. I believe I know how to, with some help from a combinatorics formula.
This is a great question that I am really enjoying working through. It feels great to work through a problem like this without outside help and without getting stuck.
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