On the integrality of some P-recursive sequences

We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences due to Klazar and Luca, and propose a unified method for analysing global boundedness and algebraicity within a broader class of holonomic sequences. The central contribution is an algorithm that determines whether all, none, or a one-dimensional family of solutions to certain second-order recurrences are globally bounded. This approach generalizes earlier ad hoc methods and applies successfully to several well-known sequences from the On-Line Encyclopedia of Integer Sequences (OEIS).