Is there any proven impossible prime gap sequences?

Question

Just out of curiosity, is there any prime gap sequences that have been proven (or suspected) to never occur ? For example, I noticed that within the first million primes, there is no prime gap of $8$ followed by a prime gap of $14$, and neither $14$ followed by $8$.

Answer 1

If two primes $> 3$ are separated by a gap of $8$, for example, they have the form $$6 k - 1, 6 (k + 1) + 1$$ for some integer $k$. The number $14$ larger than the latter is $$[6 (k + 1) + 1] + 14 = 3 (2k + 7),$$ which is composite.

Alternatively, for any integer $k$, the integers $k, k + 8, k + 22$ have different residues modulo $3$, so (exactly) one of them is divisible by $3$.