Thank you for the answer. Unfortunately the sequence does not have nice properties and I have concluded that trying to approximate it will be futile.

There are no random numbers in the sequence. The sequence, due to the size, is not knowable in practice. That is why I am looking to create increasingly better approximations. A good approximation can be used to compute mostly correct values. In theory, the entire sequence could be computed and stored, but there is not enough storage and computing power in the world.

Yes, it has many many instances of the same entry. Every value is small. It is not monotone. I would say every value is in the 0-255 range.