The rabbit sequence is the infinite sequence of binary digits obtained by iterating the substitution
system 
,
, starting with 0. It arises in
a hypothetical model of rabbit reproduction, where 
corresponds to young rabbits growing old and 
corresponds to old rabbits producing young rabbits.
The successive iterations using string rewriting
are 1, 10, 101, 10110, 10110101, 1011010110110, .... Each of these strings
is an initial segment of the next. A recurrence plot
of the infinite sequence is illustrated above.
Interpreted as binary integers and converted to decimal, the successive iterations give
1, 2, 5, 22, 181, ... (OEIS A005203, starting
at index 1), with the value after 
iterations given by the recurrence
relation
 |
for integers 
, where 
, 
, and 
is the 
th Fibonacci number with
and 
.
The real number with binary expansion 
whose binary digits form the infinite sequence
(OEIS A005614) is called the rabbit
constant.
