Recently I came across this problem :
If $a_{0} = 1$ and $a_{n} = a_{[n/2]} + a_{[n/3]} + a_{[n/6]}$ where $[x]$ denotes the greatest integer not exceeding $x$ then show that $$\lim_{n \to \infty}\dfrac{a_{n}}{n} = \frac{12}{\log 432}$$
The recurrence relation is very strange and seems to be multiplicative (has terms like $a_{[n/2]}$) instead of being additive (having terms like $a_{n - 1}$ etc) and I wonder how such a recurrence can be solved. Any suggestions are welcome!
