# Ultimately periodic sequence

A sequence over some set satisfying the condition

for all sufficiently large values of and some is called ultimately periodic with period ; if this condition actually holds for all , is called periodic (with period ). The smallest number among all periods of is called the least period of . The periods of are precisely the multiples of . Moreover, if should be periodic for some period , it is actually periodic with period .

One may characterize the ultimately periodic sequences over some field by associating an arbitrary sequence over with the formal power series

Then is ultimately periodic with period if and only if is a polynomial over . Any ultimately periodic sequence over a field is a shift register sequence. The converse is not true in general, as the Fibonacci sequence over the rationals shows (cf. Shift register sequence). However, the ultimately periodic sequences over a Galois field are precisely the shift register sequences. Periodic sequences (in particular, binary ones) with good correlation properties are important in engineering applications (cf. Correlation property for sequences).

#### References

[a1] | D. Jungnickel, "Finite fields: Structure and arithmetics" , Bibliographisches Inst. Mannheim (1993) |

**How to Cite This Entry:**

Ultimately periodic sequence.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ultimately_periodic_sequence&oldid=15942