An infinite frieze is a sequence of rows where each row is a function and the sequence satisfies:
From the definition it is clear that the whole infinite frieze is determined from the row . However, some rows may not give a rise to an infinite frieze or a frieze at all. For example if and otherwise. Therefore a question arises how many friezes there are.
By calculating the first rows of several friezes, which can be done quickly by writing a program and running it online, one can observe a general polynomial formula(thus not containing any division operation) defining a row from the previous two rows.
Let be a sequence of rows where each row is a function and the sequence satisfies the 2 conditions of an infinite frieze. Let there be a positive integer such that for all if then ,i.e. is given by a period of a length . Then .
The formula is proved by the induction. For the base case
For the inductive case assume . Then
Note that if , and , then and . This guarantees that if is given by the numbers greater than 1, then every next row will have greater entries than the preceding one and the formula from the lemma will hold as the division by 0 never occurs. This gives:
Let be given by the period of a length , i.e. . Then defines an infinite frieze .
If the row is not periodic, but its entries are integers greater than 1, then it still gives a rise to an infinite frieze for one may consider it as if defined by periods of an arbitrarily large length.
Every row gives a rise to an infinite frieze.
The number of infinite friezes is .