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.
