Patterns Occurring During the
Confrac Expansion of Quadratic Integers

Pierre Douillet

Résumé:

The confrac expansion of a quadratic integer is well known to be periodic and palindromic. When is not a square and , there are two kinds of quadratic integers related to , and it is known that their period lengths are either "roughly equal" or "roughly triple".

In a first time, we show that the usual representation of fractional linear transforms by matrices can be used to classify all the encountered behaviours into a small number of classes (the signatures). From what, we obtain a complete characterisation of the eventual pairs .

In a second time, we examine the statistical distribution of these pairs when considering all the not greater than a given square.

Résumé:

Le présent article étudie le développement en fractions continues des entiers quadratiques, et en particulier les liens entre les développements de

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