On (almost) realizable subsequences of linearly recurrent sequences

Abstract

In this note we show that if $(u_n{)}_{n⩾1}$ is a simple linearly recurrent sequence of integers whose minimal recurrence of order $k$ involves only positive coefficients that has positive initial terms, then $(M{u}_{n^s}{)}_{n⩾1}$ is the sequence of periodic point counts for some map for a suitable positive integer $M$ and $s$ any sufficiently large multiple of $k!$. This extends a result of Moss and Ward [The Fibonacci Quarterly 60 (2022), 40-47] who proved the result for the Fibonacci sequence.

9 pages