Fine Tuning
1) I am not 100% sure about equating : pi d^2 and pi b^2 to the (Im) d and b values without trying to use Archimedes Circle: Which I found those two equations need
some work ? ( The Inside and Outside Trig: sin , tan. - It's best to take S(n) and do your own relations to the Geometry which is Fairly simple.
2) The Vector Equation: The First one I derived ; is very interesting: It factors all numbers by 3; then would that factor runs out, just divide by two, and so on,
basically, this equation I derived factors I guess all Integers.. Check it out.
3) I have not tried to do a lot of things yet ... I hope everything is fairly accurate and correct.
4) This Graphics seems to be better if you " Uncheck " the actual equations so you can let it's Integrations stand out.

1) The equations are a little different
2) I made the Graphics more " vector oriented "