How can I produce an automaton accepting $\{a^{2i}bc^{2k} \mid i, k \in\mathbb{N}\}$? I am essentially confused about exactly what the $2i$ and $2k$ mean. Does that mean that the automaton only accepts strings where the number of $a$'s and $b$'s are multiples of $2$ and there is one occurrence of $b$ between $a$ and $c$? Any help is greatly appreciated, thanks.
Automaton accepting $\{a^{2i}bc^{2k} \mid i, k \in\mathbb{N}\}$
regular languages;finite automata
The language $L=\{a^{2i}bc^{2k} \mid i,k \in\mathbb{N}\}$contins all the string of the form $aaa..abcc..c$ such that the number of $a$'s is even and the number of $c$'s is even (possibly different than the number of $a$'s).To see it, choose $i,k$, say $i=1,k=2$ now $a^{2i}$ means $2i$ times $a$, so we get $aabcccc$ for the above $i,k$. For any possible $i$ and $k$ we get exactly all the strings with even number of $a$s, then a $b$, then an even number of $c$'s.