There are lots of sequences, but none are relevant
For any initial values, there will be some sequence, even one defined by a rule, that has these initial values. As a simple demonstration, if the first n terms of a sequence indexed by i are given, one can simply fit a nth degree polynomial f(i) that has those terms as its values at 1, 2, 3 etc.
However, whatever sequence or sequences this might be, it is certainly not a very obvious one. And given J.K. Rowling's admitted dislike of mathematics, it seems unlikely that she had an obscure sequence in mind when writing the number of letters.
In particular, there are some in-plot reasons for the number of letters, which seem more compelling as a justification than some abstruse sequence.
In addition, the number of letters does not even follow the sequence given, nor do we really know how many there were.
So while there are certainly many sequences that fit the number of letters, we cannot determine them from the book, nor is it likely that they were intentional.