  Pisot sequences (sequences $a_n$ with initial terms $a_0=x, a_1=y$, and
defined for $n>1$ by $a_n= \lfloor a_{n-1}^2/a_{n-2} + \frac{1}{2} \rfloor$)
often satisfy linear recurrences with constant coefficients that are valid for
all $n \geq 0$, but there are also cautionary examples where there is a linear
recurrence that is valid for an initial range of values of $n$ but fails to be
satisfied beyond that point, providing further illustrations of Richard Guy's
celebrated "Strong Law of Small Numbers". In this paper we present a decision
algorithm, fully implemented in an accompanying Maple program ({\tt
Pisot.txt}), that first searches for a putative linear recurrence and then
decides whether or not it holds for all values of $n$. We also explain why the
failures happen (in some cases the `fake' linear recurrence may be valid for
thousands of terms). We conclude by defining, and studying, higher-order
analogs of Pisot sequences, and point out that similar phenomena occur there,
albeit far less frequently. This article is dedicated to Richard K. Guy (b.
Sept. 30, 1916) on his 100th birthday.
