  A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called
realizable if there is a self-map $T:X\to X$ on a set $X$ such that $a_n$ is
equal to the number of periodic points of $T$ in $X$ of (not necessarily exact)
period $n$, for all $n\geq1$. The sequence $a$ is called almost realizable if
there exists a positive integer $m$ such that $(ma_n)_{n=1}^\infty$ is
realizable. In this article, we show that certain wide classes of integer
sequences are realizable, which contain many famous combinatorial sequences,
such as the sequences of Ap\'ery numbers of both kinds, central Delannoy
numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial
coefficients. We also show that the sequences of Catalan numbers, Motzkin
numbers, and large and small Schr\"oder numbers are not almost realizable.
