Avoidance Sequence

Let \((a_n)\) be a sequence of integers defined as \[a_0 = 0; \quad a_1 = 1; \quad a_{n+1}=\text{the next integer that shares no digits with }a_n.\] How many digits does the term \(a_{2018}\) have?

Hints:

  1. Here are a few more terms in the sequence: \[a_2=2,\ a_3=3,\ ...,\ a_9=9,\ a_{10}=10,\ a_{11}=22,\ a_{12}=30,\ a_{13}=41,\, \ldots.\]
  2. Try finding the pattern for the subsequence \((a_{8k+2})_{k \geq 3}.\)
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