Sequences of pseudo-random numbers are usually generated by recurrence formulas of the type (1). In order to increase the length $L$ of the non-periodic part of a sequence, the “perturbed” sequence (2) may be used. The asymptotic distributions (3) and (4) of $L$ and $P$ are derived from elementary probability considerations, where $P$ is the length of the period that has been formed. It follows from (5) that in that case one can expect an increase in $L$ and $P$ by the factor $\sqrt M$. A numerical example shows that such distributions may be of practical value, though $P$ can hardly be regarded as random.