How do I represent "Any string except for .... " - regex

Question

I'm trying to solve a regex where the given alphabet is Σ={a,b}
The first expression is:
L1 = {a^2n b^(3m+1) | n >= 1, m >= 0}
which means the corresponding regex is: aa(a)*b(bbb)*
What would be a regex for L2, complement of L1?
Is it right to assume L2 = "Any string except for aa(a)b(bbb)"?

Answer 1

First, in my opinion, the regex for L1 = {a^2n b^3m+1 | n>=1, m>=0}
is NOT what you gave but is: aa(aa)*b(bbb)*. The reason is that a^2n, n > 1 means that there are at least 2 a and a pair number of a.
Now, the regular expression for "Any string except for aa(aa)*b(bbb)*" is:
^(?!^aa(aa)*b(bbb)*$).*$
more details here: Regex101
Explanations
aa(a)*b(bbb)* the regex you DON'T want to match
^ represents begining of line
(?!) negative lookahead: should NOT match what's in this group
$ represents end of line
EDIT
Yes, a complement for aa(aa)*b(bbb)* is "Any string but the ones that match aa(aa)*b(bbb)*".
Now you need to find a regex that represents that with the syntax that you can use. I gave you a regex in this answer that is correct and matches "Any string but the ones that match aa(aa)*b(bbb)*", but if you want a mathematical representation following the pattern you gave for L1, you'll need to find something simpler.
Without any negative lookahead, that would be:
L2 = ^((b+.*)|((a(aa)*)?b*)|a*((bbb)*|bb(bbb)*)|(.*a+))$
Test it here at Regex101
Good luck with the mathematical representation translation...

Answer 2

The first expression is:
L1 = {a^2n b^(3m+1) | n >= 1, m >= 0}
Regex for L1 is:
^aa(?:aa)*b(?:bbb)*$
Regex demo
Input
a
b
ab
aab
abb
aaab
aabb
abbb
aaaab
aaabb
aabbb
abbbb
aaaaab
aaaabb
aaabbb
aabbbb
abbbbb
aaaaaab
aaaaabb
aaaabbb
aaabbbb
aabbbbb
abbbbbb
aaaabbbb
Matches
MATCH 1
1. [7-10] `aab`
MATCH 2
1. [30-35] `aaaab`
MATCH 3
1. [75-81] `aabbbb`
MATCH 4
1. [89-96] `aaaaaab`
MATCH 5
1. [137-145] `aaaabbbb`
Regex for L2, complement of L1
^aa(?:aa)*b(?:bbb)*$(*SKIP)(*FAIL)|^.*$
Explanation:
^aa(?:aa)*b(?:bbb)*$ matches L1
^aa(?:aa)*b(?:bbb)*$(*SKIP)(*FAIL) anything matches L1 will skip & fail
|^.*$ matches others that not matches L1
Regex demo
Matches
MATCH 1
1. [0-1] `a`
MATCH 2
1. [2-3] `b`
MATCH 3
1. [4-6] `ab`
MATCH 4
1. [11-14] `abb`
MATCH 5
1. [15-19] `aaab`
MATCH 6
1. [20-24] `aabb`
MATCH 7
1. [25-29] `abbb`
MATCH 8
1. [36-41] `aaabb`
MATCH 9
1. [42-47] `aabbb`
MATCH 10
1. [48-53] `abbbb`
MATCH 11
1. [54-60] `aaaaab`
MATCH 12
1. [61-67] `aaaabb`
MATCH 13
1. [68-74] `aaabbb`
MATCH 14
1. [82-88] `abbbbb`
MATCH 15
1. [97-104] `aaaaabb`
MATCH 16
1. [105-112] `aaaabbb`
MATCH 17
1. [113-120] `aaabbbb`
MATCH 18
1. [121-128] `aabbbbb`
MATCH 19
1. [129-136] `abbbbbb`