Avoidance Sequence

Let (an)(a_n) be a sequence of integers defined as a0=0;a1=1;an+1=the next integer that shares no digits with an.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 a2018a_{2018} have?

Hints:

  1. Here are a few more terms in the sequence: a2=2, a3=3, ..., a9=9, a10=10, a11=22, a12=30, a13=41,.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 (a8k+2)k3.(a_{8k+2})_{k \geq 3}.
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