Substitution Rules
Another way of going from Sn to Sn + 1 is to replace each symbol 0 in Sn with the pair of consecutive symbols 0, 1 in Sn + 1, and to replace each symbol 1 in Sn with the single symbol 0 in Sn + 1.
Alternatively, one can imagine directly generating the entire infinite Fibonacci word by the following process: start with a cursor pointing to the single digit 0. Then, at each step, if the cursor is pointing to a 0, append 1, 0 to the end of the word, and if the cursor is pointing to a 1, append 0 to the end of the word. In either case, complete the step by moving the cursor one position to the right.
A similar infinite word, sometimes called the rabbit sequence, is generated by a similar infinite process with a different replacement rule: whenever the cursor is pointing to a 0, append 1, and whenever the cursor is pointing to a 1, append 0, 1. The resulting sequence begins
- 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, ...
However this sequence differs from the Fibonacci word only trivially, by swapping 0's for 1's and shifting the positions by one.
A closed form expression for the so-called rabbit sequence:
The nth digit of the word is where is the golden ratio and is the floor function.
Read more about this topic: Fibonacci Word
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