$11 \cdot 44$ Let $a_{0}, a_{1}, a_{2}, \cdots$ be an increasing sequence of non-negative integers such that every non-negative integer can be uniquely expressed in the form $a_{i}+2 a_{j}+4 a_{k}$, where $i, j, k$ are not necessarily distinct. Determine $a_{1998}$.
(39th International Mathematical Olympiad Shortlist, 1998)
[Solution] Clearly, $a_{0}=0, a_{1}=1$. If there are non-negative integers $a_{0}, a_{1}, \cdots a_{n}$ that meet the conditions, and $a_{n+1}$ is the first number that cannot be expressed as $a_{i}+2 a_{j}+4 a_{k}$ $(i, j, k \in\{0,1, \cdots, n\})$, then $a_{n+1}$ is uniquely determined by $a_{0}, a_{1}, \cdots, a_{n}$. Therefore, the sequence $\left\{a_{n}\right\}$, if it exists, is unique.
Every natural number $n$ can be represented in binary as
$$
n=d_{k} \cdot 2^{k}+d_{k-1} \cdot 2^{k-1}+\cdots+d_{1} \cdot 2+d_{0},
$$

Consider the sequence
$$
a_{n}=d_{k} \cdot 8^{k}+d_{k-1} \cdot 8^{k-1}+\cdots+d_{1} \cdot 8+d_{0}.
$$

where $d_{i}(i=0,1, \cdots, k) \in\{0,1\}, k \in N \cup\{0\}$.
Since every non-negative integer $m$ can be uniquely represented in octal as:
$$
m=c_{s} \cdot 8^{s}+c_{s-1} \cdot 8^{s-1}+\cdots+c_{1} \cdot 8+c_{0},
$$

where $c_{i} \in\{0,1, \cdots, 7\}, 0 \leqslant i \leqslant s, s \in N \cup\{0\}$.
Each $c_{i} \in\{0,1, \cdots, 7\}$ can be uniquely expressed as
$$
c_{i}=d_{i}+2 d_{j}+4 d_{k}, d_{i}, d_{j}, d_{k} \in\{0,1\}
$$

Therefore, $m$ can be expressed as
$$
m=a_{i}+2 a_{j}+4 a_{k}
$$

where $a_{i}, a_{j}, a_{k}$ have the form (1), and $a_{i}, a_{j}, a_{k}$ are uniquely determined by (2).
On the other hand, if there is (3), where $a_{i}, a_{j}, a_{k}$ have the form (1), then by the uniqueness of octal representation, their coefficients are uniquely determined by (2), and thus the representation of $m$ is unique.
Therefore, (1) is the required sequence.
Since $1998=2^{10}+2^{9}+2^{8}+2^{7}+2^{6}+2^{3}+2^{2}+2^{1}$,
then $a_{1998}=8^{10}+8^{9}+8^{8}+8^{7}+8^{6}+8^{3}+8^{2}+8^{1}$.