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##

Berstein's Families

The following is a list of families of continued fractions
obtained by Berstein and published in [1]. The partial
quotients in these families are predictable, and the functions for
them are given in his paper, but to save space only the family and
its primitive period length are given. Using the notation and
variables of Levesque and Rhin in [3] (Berstein's families
are also listed here), *M* will denote a family of continued
fractions and *A*, *B*, and *C* will be as they are used in
[3]. The list below also appears in the same order and
format as it does in [3].
Let
*A* = 2*a* + 1, where *a* and *k* are positive integers. The
families associated with *A* are the following:

Let
*B* = 2^{d}(2*a* - 1) such that *a* and *d* are positive integers
with *ad* > 1. The families associated with *B* are the
following:

For the remainder of this section, let *a* and *k* be natural
numbers. Berstein obtained the following families, where
*C* =
2^{d}(2*a*-1) + 1:

Berstein obtained 12 more families, but their form and primitive
period length are not as easily stated as the families above.

** Next:** Levesque and Rhin's Families
** Up:** Families of Continued Fractions
** Previous:** Families of Continued Fractions
*mcenter*

*2000-01-06*