  The core of a finite-dimensional modular representation $M$ of a finite group
$G$ is its largest non-projective summand. We prove that the dimensions of the
cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is
Omega-algebraic, in the sense that the non-projective summands of $M^{\otimes
n}$ fall into finitely many orbits under the action of the syzygy operator
$\Omega$. Similarly, we prove that these dimension sequences are eventually
linearly recursive when $M$ is what we term $\Omega^{+}$-algebraic. This
partially answers a conjecture by Benson and Symonds. Along the way, we also
prove a number of auxiliary permanence results for linear recurrence under
operations on multi-variable sequences.
