Is there a solution/approximation for the non-linear difference equation $c\_n = c\_{n-1}+c\_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$? 



Is there a solution/approximation for the non-linear difference equation $c_n = c_{n-1}+c_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$?
As Aaron Meyerowitz mentions when $\alpha=\frac{1}{b}$ the sequence is related to the number of partitions of $bn$ into powers of $b$. The asymptotic value of this sequence was determined by de Bruijn (On Mahler's partition problem). I believe his methods can be used to get asymptotic values for general $\alpha$, though I haven't checked this carefully. By looking at the generating function you can reduce it to studying the asymptotics of the coefficients of a power series $F(x)$ (which should be approximately the generating function of your sequence) which satisfies
$$F(x)=\frac{1-x^{\lceil 1/\alpha\rceil}}{(1-x)^2}F(x^{\lceil 1/\alpha\rceil})$$
this is a Mahler type functional equation, and there are standard methods in the literature to obtain asymptotics.


