arithmetic – Counting puzzle #1: Operate combos

_{Not along with my perform optimization puzzles, additionally sorry for the extraordinarily troublesome discrete arithmetic puzzle}

In order chances are you’ll or might not know, I’ve not too long ago uploaded 2 perform optimization puzzles which have been solely a part of the inspiration for this puzzle. The remainder of the inspiration largely got here from this remark from @AxiomaticSystem:

I do know this was imagined to discuss with my minimal perform optimization puzzles, however I wish to attempt one thing right here earlier than doing that. Right here is the puzzle I’ve created right here:

Let$$start{align}f(n):=&,n^2g(n):=&,2nh(n):=&,4n+1end{align}$$and let set $mathbb S$ be the set of pos. integers that may be achieved by making use of any mixture of $f$s, $g$s, and $h$s to 0. What number of pos. integers $le2048$ are in $mathbb S$?

Issues to say

1. Partial solutions are 100% allowed and are going to be inspired on this query! It is because it is a actually troublesome drawback and I do not wish to discourage anybody from answering.
2. This took me round 1.5 hours to resolve^{1}, so I’ve been capable of confirm that there’s a solution that may be reached for this puzzle.
3. To get the $shade{inexperienced}✓$, it’s a must to present your work on how you bought the entire numbers and the ultimate reply.

Trace:

That is additionally considerably primarily based off of Blackpenredpen’s modification of a 2020 Oxford math admissions query, so this screenshot might turn out to be useful when arising with a method to resolve it (from a deleted query of mine on Arithmetic.SE that was asking if my resolution to the aforementioned drawback was appropriate):

_{^{1}with out a brute pressure algorithm (I can not program that properly), though I believe these are allowed below the foundations that I’ve acknowledged so long as each quantity in $mathbb S$ is printed and the alg. is given, though please attempt to remedy it your self first.}