Answer: 351

Solution:

Write $1$ for a house that gets mail and $0$ for one that does not. Valid patterns are binary strings of length $19$ with no two consecutive $1$s and no three consecutive $0$s. Let $A_n,$ $B_n,$ $C_n$ count valid length-$n$ strings ending in $1,$ in exactly one $0,$ and in exactly two $0$s. A $1$ may follow either kind of $0$-ending, a single $0$ may follow a $1,$ and a second $0$ may follow a single $0:$ $$\begin{aligned} A_n &= B_{n-1} + C_{n-1}, \\ B_n &= A_{n-1}, \\ C_n &= B_{n-1}. \end{aligned}$$

Starting from $A_1 = B_1 = 1,$ $C_1 = 0$ and iterating, the totals $A_n + B_n + C_n$ run $2,$ $3,$ $4,$ $5,$ $7,$ $9,$ $12,$ $16,$ $21,$ $28,$ $37,$ $49,$ $65,$ $86,$ $114,$ $151,$ $200,$ $265,$ $351.$

For $n = 19$ the count is $351.$