sequences and series - Sums like $1+2+...=-1/12$ which give $1/3$? [Not a duplicate]

This isn't a duplicate: I am looking for a sum or sums which are similar to $1+2+3+\cdots=-\frac{1}{12}$ but which have an answer (via analytic continuation) of $\pm \frac{1}{3}$ rather than $-\frac{1}{12}$. By similar I mean it is an infinite series of whole numbers.