Let X be a compact metric space and \(T: X\rightarrow X\) be a continuous map. In this paper, we define a metric similar to Bowen’s metric on X via a given sequence of non-negative integers. Using this metric, we introduce the concepts of subsequential Bowen topological entropy and subsequential packing topological entropy. Moreover, we establish a variational principle linking subsequential Bowen entropy to its measure-theoretic counterpart and two variational inequalities connecting subsequential packing entropy to two distinct measure-theoretic entropies.
