I have this language L that contains only one string: $a_{1}a_{1}a_{2}a_{1}a_{1}a_{2}a_{3}a_{1}a_{1}a_{2}a_{1}a_{1}a_{2}a_{3} ....a_{n}...a_{n}$
written more concisely $(..(a_{1}^{2}a_{2})^{2}a_{3}^{2}..)^{2}$
I am asked to find the length of the string I found it to be $2(2^{n}-1)$ and i proved it by induction.
Now, I am asked to reduce the length of this regular expression by using the intersection operation considering that the language corresponding to $R1\bigcap R2$ is the intersection of the languages corresponding to R1 and R2, respectively.
How can I get that string by intersecting languages of different sorts using those n symbols?