Integrality of a sum

Consider the following sequence defined as a sum
$$a_n=sum_{k=0}^{n-1}frac{3^{3n-3k-1},(7k+8),(3k+1)!}{2^{2n-2k},k!,(2k+3)!}.$$

QUESTION. For $ngeq1$, is the sequence of rational numbers $a_n$ always integral?

Integrality of polyhedra

  1. Given two polyhedra in $H$ representation $P_1:Axleq b$ and $P_2:Bxleq c$ which are integral are bounded when is their intersection also integral?

  2. Given two polyhedra in $H$ representation $P_1:Axleq b$ and $P_2:Bxleq c$ which are integral are bounded is there a polynomial time algorithm to find the $H$ representation of the integral polyhedron containing the integer points of their intersection?

complexity theory – Is there a recursive problem encoding the Turing integrality of a computational model?

Suppose we have a calculation model $ C $ we want to show that Turing is complete. The usual strategy would be to imitate $ C $ any computational model that we already know to be complete Turing (for example, an arbitrary Turing machine). On the other hand, a computational model is complete Turing if and only if it can compute (the indicator function of) all recursive sets. Is there a special recursive set $ S $ (or finite subset of recursive sets) so that it is sufficient to $ C $ calculate $ S $ to ensure that $ C $ Is Turing complete? Informally, $ S $ would be a set whose ramification, looping and memory calculations are all inevitably necessary. If such sets exist, what are the simplest?

In a practical sense, the usefulness of these sets would be the following test: if your computational model can solve this type of problem (eg numerical or geometric), then it's Turing complete.