In my research I have come across the need to sum simple looking geometric series-like sums. Neither Maple, nor Mathematica, nor Wolfram Alpha and not even OEIS (On-line Encyclopedia of Integer Sequences) has any answers. The simplest such sum is $$\sum_{k=0}^{n-1} a^{k^2}$$ I would also need higher powers such as $$\sum_{k=0}^{n-1} a^{k^3}$$ $$\sum_{k=0}^{n-1} a^{k^4}$$ etc. In general the sum is $$\sum_{k=0}^{n-1} a^{k^m}$$ for m = 2, 3, ... It can of course be generalized to $m$ being a general complex number but I would be happy with just positive integers larger than 1 for the moment.
Of course the interesting cases are where $a \neq 0$ or $1$.
For $m=1$ the answer is well known. It is simply the (finite) geometric series $$\sum_{k=0}^{n-1} a^k = \frac{a^n-1}{a-1} \qquad (a \neq 1)$$
This is usually stated as $$\sum_{k=0}^n a^k = \frac{a^{n+1}-1}{a-1} \qquad (a \neq 1)$$
with $n+1$ terms. The forms I use above are written to contain $n$ terms.
For the simplest numerical case we can take $a=2$, $m=2$. Then we are looking for $$\sum_{k=0}^{n-1} 2^{k^2}$$
The sequence of partial sums begins as 1, 3, 19, 531, 66067, 33620499 ... This is unknown to OEIS.
These sums look so basic that I am surprised by the lack of information about them. Does anyone have a pointer?