Mica wrote a list of numbers using the following procedure. The first number is $1$, and then, at each step, he wrote the result of adding the previous number plus $3$. The first numbers on Mica's list are $$1, 4, 7, 10, 13, 16,\dots.$$ Next, Facu underlined all the numbers in Mica's list that are greater than $10$ and less than $100000,$ and that have all their digits the same. What are the numbers that Facu underlined?
1. **Identify the sequence**: The sequence given is an arithmetic sequence where the first term \(a_1 = 1\) and the common difference \(d = 3\). The general term of the sequence can be written as:
   \[
   a_n = 1 + (n-1) \cdot 3 = 3n - 2
   \]

2. **Identify the numbers with all digits the same**: We need to find numbers in the sequence that have all digits the same and are between 10 and 100000. These numbers are of the form \(11, 22, 33, \ldots, 99999\).

3. **Check divisibility by 3**: Since the sequence \(a_n = 3n - 2\) must be equal to numbers with all digits the same, we need to check which of these numbers can be written in the form \(3n - 2\).

4. **Case 1: Two-digit numbers**:
   - Numbers with all digits the same: \(11, 22, 33, 44, 55, 66, 77, 88, 99\)
   - Check each number:
     \[
     11 = 3n - 2 \implies 3n = 13 \implies n = \frac{13}{3} \quad (\text{not an integer})
     \]
     \[
     22 = 3n - 2 \implies 3n = 24 \implies n = 8 \quad (\text{integer})
     \]
     \[
     33 = 3n - 2 \implies 3n = 35 \implies n = \frac{35}{3} \quad (\text{not an integer})
     \]
     \[
     44 = 3n - 2 \implies 3n = 46 \implies n = \frac{46}{3} \quad (\text{not an integer})
     \]
     \[
     55 = 3n - 2 \implies 3n = 57 \implies n = 19 \quad (\text{integer})
     \]
     \[
     66 = 3n - 2 \implies 3n = 68 \implies n = \frac{68}{3} \quad (\text{not an integer})
     \]
     \[
     77 = 3n - 2 \implies 3n = 79 \implies n = \frac{79}{3} \quad (\text{not an integer})
     \]
     \[
     88 = 3n - 2 \implies 3n = 90 \implies n = 30 \quad (\text{integer})
     \]
     \[
     99 = 3n - 2 \implies 3n = 101 \implies n = \frac{101}{3} \quad (\text{not an integer})
     \]
   - Valid numbers: \(22, 55, 88\)

5. **Case 2: Four-digit numbers**:
   - Numbers with all digits the same: \(1111, 2222, 3333, \ldots, 9999\)
   - Check each number:
     \[
     1111 = 3n - 2 \implies 3n = 1113 \implies n = 371 \quad (\text{integer})
     \]
     \[
     2222 = 3n - 2 \implies 3n = 2224 \implies n = 741 \quad (\text{integer})
     \]
     \[
     3333 = 3n - 2 \implies 3n = 3335 \implies n = 1112 \quad (\text{integer})
     \]
     \[
     4444 = 3n - 2 \implies 3n = 4446 \implies n = 1482 \quad (\text{integer})
     \]
     \[
     5555 = 3n - 2 \implies 3n = 5557 \implies n = 1853 \quad (\text{integer})
     \]
     \[
     6666 = 3n - 2 \implies 3n = 6668 \implies n = 2223 \quad (\text{integer})
     \]
     \[
     7777 = 3n - 2 \implies 3n = 7779 \implies n = 2594 \quad (\text{integer})
     \]
     \[
     8888 = 3n - 2 \implies 3n = 8890 \implies n = 2964 \quad (\text{integer})
     \]
     \[
     9999 = 3n - 2 \implies 3n = 10001 \implies n = 3334 \quad (\text{integer})
     \]
   - Valid numbers: \(1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999\)

6. **Case 3: Five-digit numbers**:
   - Numbers with all digits the same: \(11111, 22222, 33333, \ldots, 99999\)
   - Check each number:
     \[
     11111 = 3n - 2 \implies 3n = 11113 \implies n = 3705 \quad (\text{integer})
     \]
     \[
     22222 = 3n - 2 \implies 3n = 22224 \implies n = 7408 \quad (\text{integer})
     \]
     \[
     33333 = 3n - 2 \implies 3n = 33335 \implies n = 11112 \quad (\text{integer})
     \]
     \[
     44444 = 3n - 2 \implies 3n = 44446 \implies n = 14815 \quad (\text{integer})
     \]
     \[
     55555 = 3n - 2 \implies 3n = 55557 \implies n = 18519 \quad (\text{integer})
     \]
     \[
     66666 = 3n - 2 \implies 3n = 66668 \implies n = 22222 \quad (\text{integer})
     \]
     \[
     77777 = 3n - 2 \implies 3n = 77779 \implies n = 25926 \quad (\text{integer})
     \]
     \[
     88888 = 3n - 2 \implies 3n = 88890 \implies n = 29629 \quad (\text{integer})
     \]
     \[
     99999 = 3n - 2 \implies 3n = 100001 \implies n = 33334 \quad (\text{integer})
     \]
   - Valid numbers: \(11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999\)

Conclusion:
The numbers that Facu underlined are:
\[
\boxed{22, 55, 88, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999}
\]