jmc

We consider the opposite; we try to find the number of words that do not contain A, and then subtract it from the total possible number of words. So we have a few cases to consider:

$\bullet$ One letter words: There is only $1$ one-letter word that contains A, that's A.

$\bullet$ Two letter words: There are $19\times 19=361$ words that do not contain A. There is a total of $20\times 20=400$ words, so we have $400-361=39$ words that satisfy the condition.

$\bullet$ Three letter words: There are $19\times 19\times 19=6859$ words without A, and there are $20^3=8000$ words available. So there are $8000-6859=1141$ words that satisfy the condition.

$\bullet$ Four letter words: Using the same idea as above, we have $20^4-19^4=29679$ words satisfying the requirement.

So this gives a total of $1+39+1141+29679=\boxed{30860}$ words.