Don't make this a paper. This has been done over the centuries by so many brilliant mathematicians.
Reading Gauss, Newton, Hardy, Littlewood, Ramanujam and so many other mathematicians over the years will tell you that everything that you imagine has already been done.
You have to read many papers for many years, understand them in depth to come up with new observations using those publications as your base.
But hey... don't stop finding stuff by yourself. It is a good thing to do this kind of stuff.
regarding difference between consecutive cubes, I am more interested in explaining to myself why $d(x^3)/dx = 3x^2$ whereas for natural numbers $(n+1)^3 -n^3 = 3n^2 + 3n + 1$
all I can think of is
$lim (3n^2 + 3n + 1)$ =~ $3n^2$ as $n^2$ dwarfs $3n$.