Consider the sequence $1,-2,-1,2,1,-2,-1\ldots$ I'm trying to find the recurrence relation and finding a closed-form solution to it.
I'm stuck at trying to find how the numbers relate to each other. I've tried to construct a $2 \times 2$ square with center at the origin and vertices: $\{(-1,1), (1,1),(-1,-1),(1,-1)\}$
I found out that the coordinates, in order: $\{(1,0),(-1,-1),(-1,0),(1,1)\}$ by adding their $x$ and $y$ coordinate construct the sequence, and furthermore connecting the coordinates draws out a nice parallelogram. $\{(0,1), (-1,-1),(0,-1),(1,1)\}$ also works with simlar property.
I've tried to decompose the sequence into the difference between each term, alas yielded no results either. What's really messing me up is the interval of $2$ between the positives and negatives.
A recurrence relation of first order is really best suited to capture exponential growth and similar patterns. If you want oscillation (like we have here), it's a lot easier if you go for a second order recurrence relation. In this case: $$ a_1 = 1\\ a_2 = -2\\ a_n = -a_{n-2} $$ will do the trick.