Firstly, please excuse the informal style of my explanation, as I am not a mathematician, although I am aware that this can be explained in more formal terms.

I have mapped integers to points on a circle on the complex plane in the following way: $$ a_n=\prod _{j=1}^n (-1)^{2 (n \bmod j)/j} $$ I then took a sequence of partial sums of $a_n$: $$ b_n=\sum _{j=1}^n a_j $$ I think of it as a path made of vectors of length 1 on the complex plane. I then plotted $b_n$, and I saw this beautiful vine-like shape:

The path goes clockwise, then the spin accelerates until it turns anticlockwise and moves somewhere else. In order to find out more about the "whirlpools" and "peak flows", I checked the differences between consecutive terms of $a_n$, and obtained the following plot:

The minima/"peak flows" fall at $n = 1, 4, 11, 30, 83, 226, ...$, which appears to correspond to http://oeis.org/A078141, and to be given by $$\left\lfloor e^{n-\gamma }\right\rfloor$$ I have checked that the "whirlpools" appear to correspond to $$\left\lfloor e^{n - \gamma + 1/2}\right\rfloor$$ I think this would mean that each branch of the vine is $e$ times larger than the previous one in some way...

As for the position of the "peak flows" on the complex plane, they are: $$ 1.,0.5\, +0.866025 i,0.695702\, +1.84669 i,1.28152\, +1.03625 i,1.75407\, +1.91755 i $$

for $n = 1, 4, 11, 30, 83$.

Here is a plot of the absolute values of $b_n$:

Here is an array plot of distances between terms of $a_n$, for $n <= 100$, just for fun, really:

I have a bunch of questions:

- As per title. Is this something that has been noticed before?
- Why do the "whirlpools" fall where they do on the complex plane? What is special about those values?
- Is every term of $a_n$ unique? If so, is it possible that there exists a way of learning something about the divisors of $n$ from $a_n$ or $b_n$?