General Paperfolding Sequence
The regular paperfolding sequence corresponds to folding a strip of paper consistently in the same direction. If we allow the direction of the fold to vary at each step we obtain a more general class of sequences. Given a binary sequence (fi), we can define a general paperfolding sequence with folding instructions (fi).
For a binary word w, let w‡ denote the reverse of the complement of w. Define an operator Fa as
and then define a sequence of words depending on the (fi) by w0 = ε,
The limit w of the sequence wn is a paperfolding sequence. The regular paperfolding sequence corresponds to the folding sequence fi = 1 for all i.
If n = m·2k where m is odd then
which may be used as a definition of a paperfolding sequence.
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