%N a(1) = 1, a(2) = 2; for n>2, a(n) is the smallest positive integer not yet in the sequence which shares a digit with a(n-1) but not with a(n-2), and where a(n) contains at least one digit not in a(n-1).
%C This is the digit sequence equivalent of the Enots Wolley sequence A336957. Like that sequence to avoid the sequence halting rapidly an additional rule is placed on a(n) - it must have as least one digit not in a(n-1). This implies a(n) cannot be a repdigit as otherwise a(n+1) would not exist. If this rule is removed then the sequence terminates after five terms: 1, 2, 20, 10, 11. The next term then does not exist as it must both contain and not contain the digit 1.
%C The sequence is probably infinite as any a(n) must contain at least two distinct digits, thus a(n+2) can have at most eight distinct digits. This implies that a(n+3) can always be created using a digit in a(n+2) and a digit not in a(n+2). However the behavior of the sequence as n gets very large is unknown.
%H Scott R. Shannon, <a href="/A343927/a343927.png">Image of the first 500000 terms</a>. The green line is a(n) = n.
%e a(3) = 20 as this is the smallest unused positive integer that contains a digit in a(2) = 2 while not containing any digit in a(1) = 1.
%e a(4) = 10 as this is the smallest unused positive integer that contains a digit in a(3) = 20 while not containing any digit in a(2) = 2.
%e a(5) = 13 as this is the smallest unused positive integer that contains a digit in a(3) = 10, contains a digit not in a(3), while not containing any digit in a(3) = 20.
%Y Cf. A336957, A010785, A184992, A067581.
%A _Scott R. Shannon_, May 17 2021