What is this series of numbers called? Let's say $N$ is the initial number.
$x_0 = \frac{N}{4}$
$x_1 = \frac{N - x_0}{4}$
$x_2 = \frac{x_0 - x_1}{4}$
$x_3 = \frac{x_1 - x_2}{4}$
and so on...
What is this series called? Is there a way to generalize this for dividing by any other positive numbers besides 4? I'm really bad at math, so any help would be greatly appreciated! Thanks!
Edit 1: Oh, and also, $x_0 ... x_n$ all adds up to $N$.
 A: What you've got yourself here is a good ol' fashioned recurrence relation. The pattern is $$ x_n = -\frac{1}{4}x_{n-1} + \frac{1}{4}x_{n-2}, \quad n\geq 2 \\ x_0 = \frac{1}{4}N, \ x_1 = \frac{3}{16}N$$
Since this equation has each $x_n$ depending on as far back as $x_{n-2}$, we say this is a second order recurrence relation, and since it's in the form $x_n = \sum c_i x_i$, where $c_i$ doesn't depend on $n$, it's a linear constant coefficient equation.
From this information, it's pretty simple to solve it: just assume that there is a solution that looks like $x_n = r^n$ and figure out all the possible $r$'s, and then add them together with constant coefficients to get a solution that looks like $x_n = \sum a_i r_i^n$ (assuming the $r$'s are unique; otherwise there's some extra work to do).
The upshot of this is that the sequence is completely determined from this information, so any other claims about these terms may conflict. In particular, your claim that $\sum\limits_{n=0}^k x_n = N$ for some $k$ needs to be brought into question. With a little bit of effort, you can show that $\sum\limits_{n=0}^k x_n < \frac{1}{2}N$ for all $k$, and in fact $\sum\limits_{n=0}^\infty x_n = \frac{1}{2}N$.
There's no particular name for this equation (other than linear, constant coefficient, second order recurrence relation). However, you may draw some parallels to a particularly famous recurrence relation of the same kind, namely the Fibonacci sequence, as well as the slightly more general Lucas sequences.
Hopefully this puts you on the right track to exploring more about this sequence!
