I encountered the following recurrence when I was analyzing a problem in probability theory:
$$ a_{n}=\frac{ a_{n-1}+a_{n-2}+a_{n-3}+a_{n-4}+a_{n-5}+a_{n-6} }{6}, $$
with $a_{-5}=a_{-4}=a_{-3}=a_{-2}=a_{-1}=0$ and $a_0=1$. Now because the $a_i$ have an interpretation as probabilities (when tossing a fair die, $a_i$ is the probability the total tally will be $i$ at some point, for non-negative $i$ that is) I can prove that $a_i$ tends to $2/7$ for large $i$.
However I fail to see how to prove this fact "directly", i.e. without taking recourse to a combinatorial/probabilistic interpretation. Surely there must be a way though. Can someone help me with this?
