Give the Regular-Expression (NFA) with specific Separation Patterns Question:
Given the RE (or NFA) for the set of all strings over $\Sigma ={a,b}$ such that: a occurs the odd number of times and each pair of a are separated by exactly $2n+2,n\geq 0$ b's.
Attempt:
Followed the way we construct the set of strings such that $a$ occurs the odd number of times, I got the following solution:
$$b^*a(bb(bb)^* abb(bb)^*a)^*$$
But then I found the pattern $abbaa$ is not acceptable, as $a$ can appear multiple times consecutively. Is there any systematic way to solve the above problem?
I'll assume that no $a$s can appear together without interruption by a group of $(bb)^+$s, even if they form a group of odd $a$s. In that case
$$
a(bb)^+ (a(bb)^+ a(bb)^+)^*
$$
should suffice, where $x^+$ is defined simply as $x^+ = x x^*$.