TITLE: Asymptotics of A030283
QUESTION [6 upvotes]: I wondered about the following sequence $a_i, i \in \mathbb N$ today:
$a_1=1$
$a_n={\text{Smallest integer} > a_{n-1} \text{ that does not share any decimal digits with } a_{n-1}}$

The first few numbers in that sequence are
$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 30, 41, 50, 61, 70, 81, 90, 111, ...$
I typed the numbers into the OEIS search and sure enough, an entry came up: A030283, though with little information besides the fact that someone has thought of this sequence before.

The plot of the sequence, taken from OEIS, looks like this:

clearly suggesting asymptotically exponential behavior.

My question is this:

From the logarithmic plot, it looks as if $a_n$ is roughly $10^{n/8}$. Is this the exact asymptotic behavior of the sequence? If yes, why? If no, what is the true behavior?

REPLY [4 votes]: From the $20$th entry onward, we reach the following stable pattern, wherein the first digit follows a cycle of length $8$:
$$
2 \overbrace{0\cdots 0}^n,\\
3 \overbrace{1\cdots 1}^n,\\
4 \overbrace{0\cdots 0}^n,\\
\vdots\\
9 \overbrace{1\cdots 1}^n,\\ 2 \overbrace{0\cdots 0}^{n+1}
$$
It clearly follows that 
$$
\lim_{k \to \infty}\frac{a_{k+8}}{a_k} = 10
$$
The asymptotic nature of this sequence follows.