The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

By definition, Sum_{k=1..n} a(k) = c(n) = concatenation of the first n digits of the sequence, therefore a(n) = c(n) - c(n-1). For n > 2, this defines a(n) recursively, without the need for solving an implicit equation, as the definition might suggest. - M. F. Hasler, Feb 22 2018