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?
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.