Find all sequences $\left(a_0,a_1,a_2,a_3\right)$, where for each $k$, $0\leq k\leq3$, $a_k$ is the number of times that the number $k$ appears in the sequence $\left(a_0,a_1,a_2,a_3\right)$.
1. Let's denote the sequence as $\left(a_0, a_1, a_2, a_3\right)$. According to the problem, for each $k$ where $0 \leq k \leq 3$, $a_k$ is the number of times that the number $k$ appears in the sequence $\left(a_0, a_1, a_2, a_3\right)$.
2. This means that $a_0$ is the number of times $0$ appears in the sequence, $a_1$ is the number of times $1$ appears, $a_2$ is the number of times $2$ appears, and $a_3$ is the number of times $3$ appears.
3. Since the sequence $\left(a_0, a_1, a_2, a_3\right)$ contains exactly four elements, we have the equation:
   \[
   a_0 + a_1 + a_2 + a_3 = 4
   \]
4. Additionally, each $a_k$ must be a non-negative integer, and $a_k$ must be equal to the number of times $k$ appears in the sequence. This gives us the following constraints:
   \[
   a_0 = \text{number of times } 0 \text{ appears}
   \]
   \[
   a_1 = \text{number of times } 1 \text{ appears}
   \]
   \[
   a_2 = \text{number of times } 2 \text{ appears}
   \]
   \[
   a_3 = \text{number of times } 3 \text{ appears}
   \]
5. We need to find a sequence $\left(a_0, a_1, a_2, a_3\right)$ that satisfies these conditions. Let's analyze the possible values for each $a_k$:
   - If $a_0 = 0$, then $0$ does not appear in the sequence.
   - If $a_1 = 1$, then $1$ appears exactly once in the sequence.
   - If $a_2 = 2$, then $2$ appears exactly twice in the sequence.
   - If $a_3 = 3, then $3$ appears exactly three times in the sequence.
6. Given these constraints, the only possible sequence that satisfies all conditions is:
   \[
   \left(a_0, a_1, a_2, a_3\right) = (0, 1, 2, 3)
   \]
7. To verify, we check:
   - $0$ appears $0$ times.
   - $1$ appears $1$ time.
   - $2$ appears $2$ times.
   - $3$ appears $3$ times.
   - The sum is $0 + 1 + 2 + 3 = 6$, which is incorrect. 

8. Therefore, the initial solution is incorrect. We need to find another sequence that satisfies the conditions.

9. Let's try another approach:
   - If $a_0 = 1$, then $0$ appears exactly once in the sequence.
   - If $a_1 = 2$, then $1$ appears exactly twice in the sequence.
   - If $a_2 = 1$, then $2$ appears exactly once in the sequence.
   - If $a_3 = 0$, then $3$ does not appear in the sequence.
10. This gives us the sequence:
    \[
    \left(a_0, a_1, a_2, a_3\right) = (1, 2, 1, 0)
    \]
11. To verify, we check:
    - $0$ appears $1$ time.
    - $1$ appears $2$ times.
    - $2$ appears $1$ time.
    - $3$ appears $0$ times.
    - The sum is $1 + 2 + 1 + 0 = 4$, which is correct.

The final answer is $\boxed{(1, 2, 1, 0)}$