Initial population when i count backwards?
Your calculations are the following. Assuming non-overlapping generations, the number of ancestors you have in the last $t$ generation is given by:

$$\sum_{i=1}^t 2^t$$

This sounds correct. But there are some very strong assumptions:


Generations are non-overlapping. A more realistic model would need to consider $t$ as a continuous variable a give a probability of reproducing which depend on the age of the individual.
Never two related individuals mate together. Otherwise, you paternal grandfather might be your maternal grandfather. In reality we are all somehow related, we have a common ancestor (Hey Bro). For example if you consider that your parents are siblings and that their parents are siblings and so forth… then you don't need more than 2 great-great-great-great-grandparents. In order to keep track of such events (inbreeding), you need a stochastic model and you would need to know the population size $N(t)$ as a function of $t$. 


Tracking the change of population size through time is part of a very big in biology which is called population biology. For example you will find in this post a model of population growth of two interacting species (prey-predator)