Special numerical series with integer terms (Q2508803)

Let \(G=\{A\}\) be the set of infinite integer-valued sequences \(A=(a_1,a_2,\dots).\) The author transforms \(G\) into a ring by endowing it with operations of (componentwise) addition, subtraction, multiplication and multiplication by a scalar, and adds to these operations left shifts (with removing the left-most component) and right shifts (by adding a zero as the first term of the series). Making no further analysis of the ring he gives 13 examples of series from \(G\) whose terms can be easily expressed by simple formulas.