Decidability for repeated-root order-two linear recurrence sequences

Determine whether the first-order theory of \(\langle\mathbb{N};0,1,<,+,U\rangle\), where \(U=\{u_n:n\in\mathbb{N}\}\cap\mathbb{N}\) and \(u_n=(\alpha+\beta n)\rho^n\) is an integer order-two linear recurrence sequence with a repeated real characteristic root, is decidable, excluding cases in which \(U\) is finite or forms an arithmetic or geometric progression.

Background

Order-two integer linear recurrence sequences with a repeated real characteristic root correspond to discriminant $D=0$ and have the form $u_n=(\alpha+\beta n)\rho^n$. The paper explains that the other order-two cases are either decidable or, under the paper’s hypotheses, undecidable. The repeated-root case therefore constitutes the remaining unresolved case, with $u_n=n2^n$ given as a concrete example.