Alternating Power Fibonacci Sequence
Python 2, 30 bytes



f=lambda n:n&lt;1or f(n/4)-f(n/2)


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One-indexed.

The sequence felt like a puzzle, something that Dennis generated by having a short way to express it. The power-of-two repetitions suggest recursing by bit-shifting (floor-dividing by 2). The alternating-sign Fibonacci recursion f(n)=f(n-2)-f(n-1) can be adapted to bitshift in place of decrementing. The base case works out nicely because everything funnels to n=0.