Enzymind Labs

A list of a few of the properties of the Period-doubling sequence:

[To Do: reiterate the rules, graphical rule, and 1-D image of the sequence] 

[Also see Sloane's A035263, for a long list of properties and relationships to other sequences.]

- Aperiodic (no sub-string is repeated at a fixed interval indefinitely)

- Perfectly self-similar, in the sense that there are an infinite number of unique periodic subsets that are themselves the Period doubling sequence (see Self-similarity of the period-doubling sequence). For every integer n, starting at the nth element, the sequence formed by every nth element is the Period doubling sequence.

- Two thirds of the elements are one of the two symbols (the starting symbol), an one third are the other symbol.

- Encodes the first difference of the Thue-Morse sequence.

- The index of elements of the first (more frequent) symbol have an even number of 2's in the index's prime factorization, and the indices of the second (less frequent) have an odd number of 2's in their prime factorizations.