I came across this definition of recursive series in a textbook. It goes as follows: $$a_0 = 0$$ For all $n \in N$, $$a_{n+1} = 2a_{n}+n$$
I am not sure if this is just me being silly, but I somehow find the $a_{1}$ term to be awkward to fit into this definition since $n$ belongs to the set of $\{1,2,3,4....\}$. To get the $a_1$ term, I have to do: $$a_{0+1} = 2a_{0} + 0$$, but that formula says $n \in N$. Does my confusion make sense?
Your confusion does make sense, but so does the equation.
Depending on where you are and what academic field you're studying in, the set of natural numbers may or may not include 0 (here's a link to a wikipedia article with an ISO standard stating 0 is included: https://en.wikipedia.org/wiki/ISO_31-11). This is a matter of lack of standardization of definition.