Is there a general way/technique to evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, we consider ${a \choose b}=0$ if $b>a$)
In other words, I want the sum of the binomial coefficients ${n \choose i}$ where $i$ is less than or equal to $n$ and belongs to the arithmetic progression $r,r+k,r+2k,..., $ etc.
If $r=0$, we're looking for the sum of the binomial coefficients ${n \choose i}$ such that $k|i$, i.e., we want to evaluate the sum ${n \choose 0}+{n \choose k}+{n \choose 2k}+...$, which can be done using the roots of unity filter technique (link here) for the sum of the coefficients $a_i$ of a polynomial $f$ such that $k|i$ (in this case $f$ would be the polynomial $(1+x)^n$)
Any help? Thank you!