Solution:
Consider the elements of the sets mod 4. Then we would need to have sets of the form $\{0,0,0\}$, $\{0,2,2\}$, $\{0,1,3\}$, $\{1,1,2\}$, or $\{2,3,3\}$. In the set $\{1,2, \ldots, 19\}$ there are four elements divisible by 4 and 5 elements congruent to each of $1,2,3 \bmod 4$. Hence the desired number is given by
$$\binom{4}{3} + \binom{4}{1} \binom{5}{2} + \binom{4}{1} \binom{5}{1} \binom{5}{1} + \binom{5}{2} \binom{5}{1} + \binom{5}{1} \binom{5}{2} = 244$$