Given rational numbers $a_1,\ldots, a_k$ and $u_0, \ldots, u_k$, let $(u_n)_{n \geq k}$ be the linear recurrence defined by $$u_n := a_1 u_{n-1} + \cdots + a_k u_{n-k}, \text{ for } n \geq k .$$ Obviously, $u_n \in \mathbb{Q}$ for any $n \geq 0$.
My question is: Is there an effective way to decide if it is true that $u_n \in \mathbb{Z}$ for all sufficiently large $n$ ?
The problem is effectively decidable. To test whether $u_n$ is eventually integral, first use the recurrence relation for $u_n$ to construct relatively prime polynomials $A,B\in \mathbb{Z}[x]$ such that the rational function $A/B$ has power series expansion $\sum_nu_nx^n$. (Here relatively prime will always mean no common factor in the ring $\mathbb{Z}[x]$ besides $\pm1$.) Then the following Lemma provides an effective test on $A$ and $B$ that determines whether or not $u_n$ is eventually integral.
Lemma. Suppose that $A,B\in \mathbb{Z}[x]$ are relatively prime, and that $A/B$ has power series expansion $\sum_{n\ge0}u_nx^n$. Let $B=cC$, where $c$ is the gcd of the coefficients of $B$. Then the sequence $u_n$ is eventually integral if and only if
Proof of the Lemma. The if direction: we assume that Conditions 1 and 2 hold, and prove that the sequence $u_k$ is eventually integral.
By Condition 2, $$A=cD+CE,$$ for some choice of $D,E\in \mathbb{Z}[x].$ Dividing by $B$, and using $B=cC$, $$\tag{*}\dfrac{A}{B}=\dfrac{D}{C}+\dfrac{E}{c}.$$ But Condition 1 implies that $C$ has the form $\pm(1-xC_1)$, for some $C_1\in \mathbb{Z}[x]$. Therefore $D/C$ has the form $$\pm D(1+(xC_1)+(xC_1)^2+\ldots).$$ It follows that the power series for $D/C$ has all integral coefficients. Since $E/c$ is a polynomial with rational coefficients, it follows from ($*$) that the power series $\sum u_kx^k$ for $A/B$ eventually has integral coefficients.
The only if direction: We assume that the sequence $u_k$ is eventually integral, and verify Conditions 1 and 2.
Remark. At this point it will be convenient to extend the usual notion of the content of a polynomial to power series $f=\sum_nu_nx^n$ with eventually integral coefficients. Define $$\gamma(f)=\prod_{p \text{ prime}}p^{\min_n(v_p(u_n))},$$ where $v_p(u_n)$ is the exponent to which $p$ appears in the rational number $u_n$. If $P\in \mathbb{Z}[x]$ then the product $Pf$ again has eventually integral coefficients, and it holds that $\gamma(Pf)=\gamma(P)\gamma(f)$. The proof is similar to the case of two polynomials.
Proof of Condition 1. Since $A$ and $B$ are relatively prime, there are polynomials $U,V\in \mathbb{Z}[x]$ and an integer $m\ne0$ such that $AU+BV=m$. Let $f=\sum_nu_nx^n$ be the power series expansion of $A/B$. By factoring out $B$, write the equation $AU+BV=m$ in the form $$\tag{**}B(fU+V)=m.$$ Since $c=\gamma(B)$, the multiplicativity of the content function $\gamma$ implies that $$c\gamma(fU+V)=m.$$ But ($**$) implies that $B(0)t=m$, where $t$ is the constant term of $fU+V$. Therefore $B(0)t=c\gamma(fU+V)$, or equivalently $$ \dfrac{B(0)}{c}\cdot \dfrac{t}{\gamma(fU+V)}=1.$$ Since the two factors are both integers, it follows that $B(0)=\pm c$. This proves Condition 1.
Proof of Condition 2. As before, let $f$ denote the power series for $A/B$. We note that the power series $cf$ has all integer coefficients. Indeed, taking the content of both sides of the equation $A=Bf$, we have $$\gamma(A)=c\gamma(f)=\gamma(cf).$$ The equation implies that $cf$ has integer content. Therefore all coefficients of $cf$ are integers.
It follows that $f$ has the form $D/c+g$, for some $D\in \mathbb{Z}[x]$ and some power series $g$ with integer coefficients. Multiplying the equation $A/B=D/c+g$ by $B$ and using the definition $B=cC$, we conclude that $$A=CD+c\cdot(Cg).$$ But $Cg$ is visibly a polynomial, being a difference of two polynomials. Therefore $A$ is an element of the ideal of $\mathbb{Z}[x]$ generated by $c$ and $C$. This completes the proof of the Lemma.
Notes.
