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Pisot sequences (sequences $a_n$ with initial terms $a_0=x, a_1=y$, anddefined 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 forall $n \geq 0$, but there are also cautionary examples where there is a linearrecurrence that is valid for an initial range of values of $n$ but fails to besatisfied beyond that point, providing further illustrations of Richard Guy'scelebrated "Strong Law of Small Numbers". In this paper we present a decisionalgorithm, fully implemented in an accompanying Maple program ({\ttPisot.txt}), that first searches for a putative linear recurrence and thendecides whether or not it holds for all values of $n$. We also explain why thefailures happen (in some cases the `fake' linear recurrence may be valid forthousands of terms). We conclude by defining, and studying, higher-orderanalogs 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.

Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences