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Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred to as D-finite sequences. A subclass are C-finite sequences satisfying a linear recurrence with constant coefficients.We investigate the set of sequences which satisfy linear recurrence equations with coefficients that are C-finite sequences. These sequences are a natural generalization of holonomic sequences. In this paper, we show that C 2 -finite sequences form a difference ring and provide methods to compute in this ring. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; • Mathematics of computing → Combinatorial algorithms.

A sequence is called C-finite, if it satisfies a linear recurrence with constant coefficients and holonomic, if it satisfies a linear recurrence with polynomial coefficients. The class of C 2 -finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients whose leading coefficient has no zero terms. Recently, we investigated computational properties of C 2 -finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring.From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple C 2 -finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; • Mathematics of computing → Combinatorial algorithms.

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A sequence is called C-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with C-finite coefficients. Recently, it was shown that such C 2 -finite sequences satisfy similar closure properties as C-finite sequences. In particular, they form a difference ring.In this paper we present new techniques for performing these closure properties of C 2 -finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations.The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.