Count the binary strings that can be generated by alternatingly adding zeros and ones to either end; see https://math.stackexchange.com/questions/3706460.

Question3706460.java

import java.util.HashSet;
import java.util.Set;

public class Question3706460 {
    public static void main(String [] args) {
        Set<Long> strings = new HashSet<>();
        strings.add(1L);
        long bit = 2;
        boolean one = false;
        for (int n = 1;;n++,bit <<= 1,one = !one) {
            System.out.println(n + " : " + strings.size() + " / " + Math.pow(strings.size(), 1. / n));
            Set<Long> newStrings = new HashSet<>();
            if (one)
                for (Long s : strings) {
                    newStrings.add(s | bit);
                    newStrings.add((s << 1) | 1);
                }
            else
                for (Long s : strings) {
                    newStrings.add(s);
                    newStrings.add(s << 1);
                }
            strings = newStrings;
        }
    }
}