Five-letter strings with no repeats side by side

Using letters from $\{a,b,c,d,e\}$, how many $5$-letter strings can be formed in which no letter occurs more than twice and no two neighbouring letters are equal?

$1200$. Without the 'at most twice' rule there are $5\cdot4^4=1280$ strings with no equal neighbours. A letter can appear $3$ times without touching only in the pattern $X\_X\_X$, giving $5\cdot4\cdot4=80$ strings to remove, so $1280-80=1200$.