Here is the case $\alpha=1/2.$ It shows conjectures but few reults.

Here is the case $\alpha=1/2.$ with rounding down. It shows conjectures but few results. But you asked about rounding up. That is here and it seems that more is known in this $\alpha=1/2$ case including a good estimte of the growth rate. The $nth$ term is the number of partitions of $2n$ into powers of 2.