frequency spectrum - Why does a gold sequence have $2^m + 1$ symbols while a single m-sequence has only $2^m - 1$

As explained in here:

Gold sequences help generate more sequences out of a pair of m-sequences giving now many more different sequences to have multiple users. Gold sequences are based on preferred pairs m-sequences.

For example, take the polynomials $x^5+x^2+1$ and $x^5+x^4+x^3+x^2+x+1$:

By combining two of these sequences, we can obtain up to 31 plus the two m-sequences themselves, generate 33 sequences.

I do understand why there are 31 ($2^5 - 1$) sequence length for a single m-sequence, but I am not sure why there are $2^5 + 1$ for the gold sequence.

Answer

You have the preferred pair, each of length $2^m-1$. The other sequences are generated by modulo-2 addition of one of these sequences with cyclical shifts of the other sequence. There are $2^m-1$ possible shifts, so you get $2^m-1$ new sequences from the modulo-2 additions (one for each possible shift). And you still have the two original sequences of the preferred pair giving a total of $2^m-1+2=2^m+1$ sequences.