Abstract
This paper studies divisibility properties of sequences defined inductively by a 1 = 1, a n+1 = sa n + t⌊θ a m ⌋, where s,t are integers, and θ is a quadratic irrationality. Under appropriate hypotheses (especially that s + tθ be a PV-number) it is proved that the highest power of Δ that divides a n , where Δ is the discriminant of θ, tends to infinity. This is noteworthy in that truncation would normally be expected to destroy any simple algebraic structure. Moreover, we establish related results that imply the a n are not uniformly distributed modulo Δ in cases where the smaller conjugate of s + tθ exceeds 1 in modulus (the non-PV case).
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