Let 
be a sequence over a finite
alphabet 
(all the entries are elements of 
).
Define the block growth function 
of a sequence to be the number of admissible
words of length 
. For example, in the sequence
, the following words are admissible
| length | admissible words |
| 1 |  |
| 2 |  |
| 3 |  |
| 4 |  |
so 
, 
, 
, 
, and so on. Notice that
, so the block growth function is always
nondecreasing. This is because any admissible word
of length 
can be extended rightwards to produce an admissible
word of length 
. Moreover, suppose 
for some 
. Then each admissible word of
length 
extends to a unique admissible word of length
.
For a sequence in which each substring of length 
uniquely determines the next symbol in
the sequence, there are only finitely many strings of
length 
, so the process must eventually cycle and the sequence
must be eventually periodic. This gives us the following theorems:
1. If the sequence is eventually periodic, with least period 
, then 
is strictly increasing until it reaches 
, and 
is constant thereafter.
2. If the sequence is not eventually periodic, then 
is strictly increasing and so 
for all 
. If a sequence has the
property that 
for all 
, then it is said to have minimal block growth, and
the sequence is called a Sturmian
sequence.
The block growth is also called the growth function or the sequence complexity of a sequence.
