23. Find the number of ways to pave a $1 \times 10$ block with tiles of sizes $1 \times 1,1 \times 2$ and $1 \times 4$, assuming tiles of the same size are indistinguishable. (For example, the following are two distinct ways of using two tiles of size $1 \times 1$, two tiles of size $1 \times 2$ and one tile of size $1 \times 4$.

It is not necessary to use all the three kinds of tiles.)
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23. Answer: 169.

Let $a_{n}$ be the number of ways to pave a block of $1 \times n$. Then $a_{n}=a_{n-1}+a_{n-2}+a_{n-4}$ with initial conditions $a_{1}=1, a_{2}=2, a_{3}=3$ and $a_{4}=6$. Then
$$
\begin{array}{ll}
a_{5}=a_{4}+a_{3}+a_{1}=10, & a_{6}=a_{5}+a_{4}+a_{2}=18, \\
a_{7}=a_{6}+a_{5}+a_{3}=31, & a_{8}=a_{7}+a_{6}+a_{4}=55, \\
a_{9}=a_{8}+a_{7}+a_{5}=96, & a_{10}=a_{9}+a_{8}+a_{6}=169 .
\end{array}
$$