Consider how the interleaved-bits curve (perfect outer unshuffle) maps input indices to output indices for small values of n (e.g., n=3). What pattern emerges in the binary representations of these indices?

For n=3, list all 8 input indices (0 to 7) and their corresponding output indices after applying the perfect outer unshuffle. Observe how the bits are rearranged and identify the general rule for bit permutation.

Generalize the pattern observed for n=3 to arbitrary n. Derive a formula or bitwise operation that describes how each bit in the input index contributes to the output index in the interleaved-bits curve.