Solution:

Suppose the sequence has length $s$ and first term $x$. Then $$345=s\left(x+\frac{s-1}{2}\right),$$ or $$s(2x+s-1)=690.$$

Thus $s$ must be a divisor of $690$, with $x=\frac{690/s-s+1}{2}$ a positive integer. Checking the possible divisor lengths gives $$s=2,3,5,6,10,15,23.$$ These are the only lengths that keep $x$ positive and integral.

Therefore there are $7$ representations.

Thus, the correct answer is E.