TITLE: A recurrence that wiggles?
QUESTION [15 upvotes]: Consider the following sequence $a_n$:
$a_1 = 0$
$a_n = 1 + \frac{1}{2^n-2} \sum_{i=1}^{n-1} \binom{n}{i} a_i$
The first few terms are $0,1,\frac{3}{2},\frac{13}{7},\frac{15}{7}$.
The sequence comes out of the analysis of a certain process, whose asymptotics we expect to be about $\log_2 n$. Numerical computations show that indeed $a_n \approx \log_2 n - C$, where $C \approx 0.318$. It should be easy to show that $a_n = \log_2 n + O(1)$.
 Is it true that $a_n = \log_2 n - C + o(1)$ for some constant $C$? 
There are some sequences defined by similar recurrence relations for which the "constant" part actually wiggles (unfortunately the reference escapes me).

REPLY [4 votes]: Noam Elkies gives an example here.