This is a simplified coin-changing question.
Suppose the only coins available are all powers of $10$ dollars.
How many ways are there to make change for $\$ 1000000$?
In general, to make change for any amount \$$N$, the answer is $C(N)$, where
$$C(N)=C(N-10)+C(\lfloor N/10 \rfloor )$$.
That is because the change may either include at least ten \$1 coins,
and there are $C(N-10)$ solutions; or fewer than ten \$1 coins, and the remaining coins are all multiples of ten, and there are $C(\lfloor N/10 \rfloor )$ solutions.
My question is: What are the asymptotics of this recursion, as $N$ grows large?
[EDIT]
Thankyou for the search tip. I found it in a 1948 paper in Indagationes Math X "On Mahler's Partition Problem". NG De Bruijn cited Karl Mahler's 1940 paper from Journ. London Math Soc. "On a special functional equation". His formula, for base $10=r$ and number $N=rh$ was $$ \log C(rh)=\frac{1}{2\log r}\left(\log\frac{h}{\log h}\right)^2 +\left(\frac{1}{2}+\frac{1}{\log r}+\frac{\log\log r}{\log r}\right)\log h\\-\left(1+\frac{\log\log r}{\log r}\right)\log\log h+O(1)$$
De Bruijn went on to calculate the $O(1)$ which was a periodic but not differentiable function.
Well, that's sorted me out.
