On other two representations of the C-recursive integer sequences by terms in modular arithmetic

An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are integers. While in the paper “On the representation of C-recursive integer sequences by arithmetic terms” by Prunescu and Sauras-Altuzarra, the terms consist of the remainder operation, applied on a division; the representations shown here are a division applied to a remainder operation, respectively the composition of two remainder operations.